IN  MEMORIAM 
FLOR1AN  CAJORI 


TUB 


WESTERN  CALCULATOR, 


NEW  AND  COMPENDIOUS  SYSTEM 


PRACTICAL  ARITHMETIC; 

CONTAINING 

THE  ELEMENT  \RY  PRINCIPLES  AND  RULES  OP  CALCULATION  IN 
WHOLE,  MIXED,  AND  DECIMAL  NUMBERS, 

ARRANGED,  DEFINED,  AND  ILLUSTRATED, 

IN  A  PLAIN  AND  NATURAL  ORDER  5 

ADAI'TEP  TO   THE   USE  OF  SCHOOLS,   THROUGHOUT  THE   WESTERN    COUNTRY 
AND  PRESENT   COMMERCE   OF  THE   UNITED  STATES. 

% 

IN  EIGHT  PARTS. 


BY   J.  STOCKTON/ A.M. 


PITTSBURGH : 

PRINTED  AND  PUBLISHED  BY  JOHNSTON  &  STOCKTON 
MARKET  STREET. 


STEREOTYPED   BY    J.    HOWE- 

1880. 


Entered  according  to  IK  af  the  Congress,  in  the  year  1832,  bv 
JOCNSTON  &  STOCKTWR,  •  jfr  Clerk's  office  of  the  District  Court  of 
the  Western  District  W  '  . 


PREFACE. 


AMONG  the  many  systems  of  Arithmetic  now  used  in  our  American 
schools,  though  each  has  its  individual  merit,  yet  all  contain  many 
things  which  are  either  entirely  useless,  or  *of  but  little  value  to  most 
beginners. 

It  is  to  be  regretted  also,  that  in  most  of  these  systems,  even  if? 
those  parts  which  are  valuable  and  important,  the  authors  appear  not 
to  have  been  sufficiently  aware  of  giving  a  plain  and  natural  arrange, 
ment  and  system  to  the  whole.     The;e  is  not  that  visible  connexion  * 
/  between  the  parts,  which  enables  the  attentive  pupil  to  discover,  as  he    ] 
/    progresses,  that  he  is  learning  a  system,  and  not  a  number  of  separate    * 
^  and  unconnected  rules. 

In  many  things,  also,  more  attention  has  been  given  to  gratify  the 
Jr^quirifjff  of  tliR  jirn/ffipMt^thnn  to  furnish  plain,  but  necessary  instruc- 
tion to  the  beginner.  The  age,  capacity,  and  progress  of  the  scholar 
are  also  overlooked  ;  and  a  mode  ^f  instruction  _too_  learned^  and  too 
elaborate,  is  pursued.  It  is  forgotten  how  difficult  even  the  most  sim- 
ple parts  are  to  a  young  mind  ;  nor  are  the  instances  few,  in  which 
even  the  variety~of  ways  laid  down,  in  which  the  same  question  may  be 
solved,  leaves  the  learner  perplexed,  and  swells  the  size  of  the  work. 

To  remedy,  in  some  measure,  these  defects,  and  to  furnish  our  nu- 
merous schools,  in  the  western  country,  with  a  plain  and  practical  trea- 
tise of  Arithmetic,  compiled  and  printed  among  ourselves,  thereby 
saving  a  heavy  annual  expense  in  the  purchase  of  such  books,  east  of 
the  rnnnnt?iinsT  and  {ikewi^  tliP  ./..arringp  tbageof^liave  been  the  mo- 
lives  which  induced  the  compiler  to  undertake  this  work. 

In  it  the  following  objects  have  been  steadily  kept  in  view : 

1st.  Plainness  and  simplicity  of  style,  so  that  nothing  should  be  in- 
troduced above  the  common"  capacities  of  scholars,  at  the  early  age  in 
which  they  are  generally  put  to  the  study  of  Arithmetic. 

2d.  A  natural  and  lucid  arrangement  of  the  whole,  as  a  systelii,  in 
which  the  connexion  and  dependence  of  all  the  parts  may  be  easily 
discovered  and  understood.  To  accomplish  this  object,  the  work  ia 
divided  into  eight  parts,  following  each  other,  in  what  appears  to  tnc 
compiler  the  natural  and  simple  divisions  of  the  science.  Each  of 
these  parts  is  again  divided  into  sections,  following  the  same  connected 
arrangement.  In  each  of  these  sections,  the  rules  are  expressed  in  a 


M305992 


IV  PREFACE. 

short  and  plain  manner,  and  each  rulo  is  illustrated,  with  a  few  easy 
and  familiar  examples,  gradually  proceeding  from  that  which  is  sim- 
pie,  to  such  as  are  more  abstruse  and  difficult. 

3d.  Clearness  and  precision  in  the  definitions,  directions,  and  exam- 
ples. Carefully  explaining  every  technical  term  when  first  used,  and 
thereby  guarding  against  ambiguity  and  uncertainty. 

4th.  Brevity  in  each  part,  so  that  every  thing  useless,  or  unimport- 
ant, may  be  excluded ;  in  order  that  the  work  may  find  its  way  into 
schools  at  the  cheapest  rate  ;  that  parents,  when  examining  the  school- 
books  of  their  children,  may  not  find,  whilst  one  part  is  worn  out,  the 
other  is  untouched,  and  half  the  price  of  the  book  entirely  lost. 

How  far  these  objects  have  been  obtained,  must  be  left  to  the  deci- 
'  sion  of  time.  Should  the  work  be  found  to  aid  the  progress  of  scholars, 
in  acquiring  a  practical  knowledge  of  this  useful  science — to  save  ex- 
penses in  a  book  so  many  of  which  are  required ;  and  be  found  a  use- 
ful assistant  to  merchants,  mechanics,  and  farmers,  as  well  as  in  some 
degree  to  lessen  the  labor  of  teachers  in  this  branch  of  the  sciences ; 
the  compiler  will  have  obtained  his  object. 


NOTICE  TO  THE  FOURTH  EDITION. 

THE  favorable  reception,  and  wide  circulation,  of  the  former  edi- 
tions of  the  Western  Calculator,  stimulate  the  author  to  make  it  still 
more  deserving  of  public  patronage. 

He  has,  therefore,  at  the  suggestion  of  several  respectable  teachers, 
given  sundry  additional  questions  to  some  of  the  rules ;  and  also  some 
other  alterations,  which  several  years'  experience  in  teaching  has 
pointed  out. 

The  greatest  care  has  been  taken  to  prevent  errors  from  appearing. 
in  this  edition. 

Pittsburgh,  February  1,  1823. 


APHORISMS 

FOR    THE 

SCHOLAR'S  CAREFUL  CONSIDERATION  AND  ATTENTION. 


KNOWLEDGE  is  the  chief  distinction  between  wise  men 
and  fools  ;  between  the  philosopher  and  the  savage. 

The  common  and  necessary  transactions  of  business  can- 
not be  conducted  with  profit  or  honesty,  without  the  know- 
ledge of  Arithmetic. 

He  who  is  ignorant  of  this  science  must  often  be  the 
dupe  of  knaves,  and  pay  dear  for  his  ignorance. 

Banish  from  your  mind,  idleness  and  sloth,  frivolity  and 
trifling ;  they  are  the  great  enemies  of  improvement. 

Make  study  your  inclination  and  delight ;  set  your  hearts 
upon  knowledge. 

Accustom  your  mind  to  investigation  and  reflection ;  de- 
termine to  understand  every  thing  as  you  go  along. 

Commit  every  rule  accurately  to  memory,  and  never  resP 
satisfied  until  you  can  apply  it. 

As  much  as  possible  do  every  thing  yourself;  one  thing 
found  out  by  your  own  study,  will  be  of  more  real  use  than 
twenty  told  you  by  your  teacher. 

Be  not  discouraged  by  seeming  difficulties ;  patience  and 
application  will  make  them  plain. 

Endeavor  to  be  always  the  best  scholar  in  your  class, 
and  to  have  the  fewest  mistakes,  or  blots,  in'  your  book. 

"  The  wise  shall  inherit  honor,  but  shame  shall  be  the 
promotion  of  fools." 

A2 


EXPLANATION 

OF    THE 

SEVERAL  CHARACTERS  EMPLOYED  FOR  THE  SAKE  OF 
BREVITY,  IN  THIS  TREATISE. 


—  Two  parallel  lines,  signifying  equality:  as,  100  cents= 

1  dollar ;  that  is,  100  cents  are  equal  to  1  dollar. 

-f- Signifying  more,  or  addition:  as,  6  +  4=10;  that  is,  6 
arid  4  added  make  10.  This  character  is  called  Plus. 

— A  single  line,  signifying  less,  or  subtraction  :  as,  6 — 4= 

2  ;  that  is,  6  le^s  4  is  equal  to  two.     This  character  is 
called  Minus. 

x  Signifying  Multiplication:  as,  2X  4=8  ;  that  is,  2  mul- 
tiplied by  4  is  equal  to  8. 

-I- Signifying  Division:  as,  6-f-3=2  ;  that  is,  6  divided  by 

3  is  equal  to  2. 

::  :  Signifying  Proportion  :  as,  2  :  4  ::  6  :  12  ;  that  is,  as 
2  is  to  4,  so  is  6  to  12  ;  or,  that  there  is  the  same  propor- 
tion between  6  and  12,  as  there  is  between  2  and  4. 

v/  or  V  Signifying  the  square  root  of  the  number  before 
which  it  is  placed  :  as,  \/64=8  ;  that  is,  the  square  root 
of  64  is  8. 

<X  Signifying  the  cube  root :  as,  y/64=4  ;  that  is,  the  cube 
root  of  64  is  4. 

A  Vinculum,  or  chain  :  denoting  the  several  quanti- 
ties over  which  it  is  placed,  are  to  be  considered  as  one 
simple  quantitv 


THE 

WESTERN  CALCULATOR* 


PART  I 
ARITHMETIC  IN  WHOLE  NUMBERS. 


ARITHMETIC  is  the  art,  or  science,  of  computing  by  num- 
bers, and  is  generally  divided  into  five  j^incipal  parts,  or 
primary  rules  :  viz.  Numeration,  Additioi*  Subtraction, 
Multiplication,  and  Division. 


SECTION  1. 

OF  NUMERATION. 


NUMERATION  (or,  as  it  is  often  called,  Notation)  is  the  art 
of  expressing  any  given  or  supposed  number,  by  the  ten 
following  characters  :  0,  1,  2,  3,  4,  5,  6,  7,  8,  9.  The 
first  of  these  is  called  a  cipher,  the  rest  are  termed  digits  or 
figures. 

These  nine  digits  are  divided  into  three  periods,  three  in 
each  period.  The  first  period  includes  units,  tens,  and  hun- 
dreds. The  second  period  includes  thousands,  tens  of  thou- 
sands, hundreds  of  thousands.  The  third  period  includes 
millions,  tens  of  millions,  hundreds  of  millions. 

Note. — The  cipher  is  also  called  nought  and  zero.  They  are  all 
Arabic  characters. 


NUMERATION. 

The  relative  value  of  each  period,  and  the  different  fig- 
ures ID  each  period,  may  be  learned  from  the  following 

TABLE. 

3d  period.     2d  period.     1st  period. 


9 
9 
9 

8 

8 

7, 

9 

8 

7, 

6 

9 

8 

7, 

6 

5 

9 

8 

7, 

6 

5 

4, 

9 

8 

7, 

6 

5 

4, 

3 

9 

8 

7, 

6 

5 

4, 

3     2 

9 

8 

7, 

6 

5 

4, 

321, 

After  the  foregoing  table,  with  the  preceding  definitions 
and  explanations,  are  well  explained  by  the  teacher,  and 
accurately  committed  to  memory  by  the  pupil,  let  him  next 
proceed 

To  write  Numbers, 

Observing  carefully  the  following 

RULE. 

Write  down  first,  the  given  sum,  in  such  figures  as  ex- 
press  its  value,  and  then  supply  the  deficiencies  therein  with 
ciphers 

Application. 

Write  down  in  figures  the  following  numbers. 

1.  Sixteen. 

2.  Forty -nine. 


ADDITION.  9 

3.  Three  hundred  and  eighty-five. 

4.  Two  thousand  six  hundred  and  ten. 

5.  Sixty-four  thousand,  five  hundred  and  thirty-six. 

6.  Two  hundred  and  fifty-three  thousand,  eight  hundred 
and  forty-two. 

7.  Five  millions,  six  hundred  thousand  and  six. 

8.  Ninety  millions,  three  hundred  and  five. 

9.  Eight  hundred  and  twenty-nine  millions,  six  thousand 
and  two. 

Write  dcnvn  in  words  at  length  the  following  sums : 
5,  17,  35,  458,  6829,  72348,  384721,  2683200,  50678024. 

Numbers  are  also  expressed  by  letters,  and  are  called 
numeral  letters,  or  Roman  numbers.  Thus, 

12.3456       7         8        9     10    11     20       30 
1,    II,  III,  IV,  V,  VI,  VII,  VIII,  IX,  X,  XI,  XX,  XXX, 

40  41  50  60   70   80    90   100  200  500  1000 
XL,  XLI,  L,  LX,  LXX,  LXXX,  XC,  C,  CC,  D,  IVL 

When  a  letter  of  less  value  stands  before  one  of  a  greater, 
it  diminishes,  but  when  placed  after,  it  increases,  the  value 
of  the  greater. 


SECTION  2. 

OF  ADDITION. 

ADDITION  is  of  two  kinds,  viz.  Simple  and  compound. 

Simple  addition  teacheth  to  collect  two  or  more  numbers 
of  the  same  denomination  into  one  sum  :  as,  6  dollars  and 
4  dollars,  make  10  dollars. 

RULE  1.  Write  the  different  numbers  in  the  given  sum  in 
such  a  manner,  that  the  units  may  stand  under  units,  tho 
tens  under  tens,  the  hundreds  under  hundreds,  &c.  till  the 
whole  is  set  down. 

2.  Begin  with  the  column  of  units,  and  add  it  into  OIK 
sum,  carrying  to  the  next  column,  one  for  every  ten,  and 
set  down  the  remainder  directly  underneath  ;  proceed  in  the 
same  manner  fnnn  tens  to  hundreds,  &c.  till  all  is  finished. 


10  ADDITION. 

Prove  the  work   by  reckoning  downwards  as  well  as  up- 
wards, and  if  the  amounts  be  equal  the  work  is  right. 


EXAMPLES. 


Dollars 
2465 
4832 
6143 


Yards 

2.  468256 
348928 
764182 


13440  added  upwards. 


13440  added  downwards. 


Feet 

3.  647502434 
861948260 
959276398 


4.  258335091     5. 

237680923 

6.  919283746 

138097653 

423315687 

213536978 

573217809 

098172635 

321325687 

532458976 

523516533 

978562313 

532175633 

653213563 

321897553 

249753290 

327865309 

213587921 

7.   5643218624 

8.       4 

9.  6856789436 

135940536 

45 

40590428 

42006302 

456 

36491 

9580469 

4567 

2849653210 

550214 

45678 

540 

32651 

456789 

34906 

4168 

4567890 

3458000 

324 

45678901 

300 

68 

456789012 

9 

4 

4567890123 

35 

10.  3683678048934 

11. 

595536210486 

2864948946496 

376891345613 

8498649476828 

765248567446 

3646280568245 

684720476828 

6421424H78427 

852134567812 

3678156496862 

236889634567 

76545S4964859 

335678902345 

SUBTRACTION.  11 

Application. 

1.  Add  125-f  23  + 16  -f  2060  +  8009574  +  6. 

^Iws.  8011804. 

2.  Add  one  hundred  and  twenty-nine,  six  hundred  and 
fifty-four,  eight  thousand  and  seventy,  ten  thousand,  and 
four  millions.  Ans.  4018853. 

3.  If  I  have  received  125  dollars  from  A,  286  from  B, 
29  from  C,  672  from  D  ;  how  much  have  I  received  from 
all  four?  Ans.  1112. 

4.  Bought  60  barrels  of  flour  from  one  man  for  480  dol- 
lars, 75  barrels  from  another  for  675  dollars,  220  from  an- 
other for  2200  dollars,  and  126  from  another  for  1386  dol- 
lars ;  how  many  barrels  of  flour  had  I,  and  how  much  did 
they  cost  me  ?  Ans.  481  barrels,  and  cost  4741  dolls. 

5.  A  farmer  raised  in  one  year  297  bushels  of  wheat, 
125  of  rye,  754  of  corn,  127  of  barley,  and  245  of  oats  ; 
how  many  bushels  did  he  raise  in  all  ?  Ans.  1548. 

6.  Add  one  thousand  two  hundred  and  nine,  four  hun- 
dred and  seventy-six,  eight  thousand  and  seventeen,  three 
millions,  one  hundred  and  nineteen  thousand,  two  hundred 
and  twenty-one  together.  Ans.  3128923. 

7.  James  was  born  in  the  year  1811  ;  in  what  year  will 
be  Le  21  years  old?  Ans.  1832. 

8.  A  father  bequeathed  to  his  5  sons  the  following  sums, 
viz :  to  George  he  gave  3560  dollars,  to  William  3240,  to 
Samuel  2850,  to  Henry  2555,  and  to  Thomas  2226  ;  how 
much  did  he  bequeath  in  all  ?  Ans.  14431  dolls. 


SECTION  3. 
OF  SUBTRACTION. 

SUBTRACTION  is  either  simple  or  compound. 
Simple  subtraction  is  the  taking  a  less  number  from  a 
greater,  and  thereby  finding  the  difference. 

RULE. 

Place  the  less  number  under  the  greater,  with  units  under 
units,  tens  undei  tens,  &c. ;  begin  with  the  units,  and  take 
the  under  figure  rrom  the  upper,  and  then  proceed  with  t  ic 


12  SUBTRACTION. 

tens,  &c.  in  the  same  manner.  But  if  the  under  figure  is 
the  greatest,  then  suppose  ten  added  to  the  upper  figure,  and 
take  the  lower  from  that  number,  carrying  1  to  the  next 
place.  Or,  take  the  lower  figure  from  10  and  add  the  upper 
one  to  the  remainder. 

PROOF. 

Add  the  remainder  to  the  less  number,  and  that  will  equal 
the  greater. 


From  446875296 
Take  234521173 


Rem.  212354123 


Proof  446875296 

From  76542189768 
Take  32478127130 


EXAMPLES. 


From  86250732493, 
Take  37014921872 


Rem.  49235610621 


Proof  86250732493 

From  5417630912 
Take       27096470 


From  90621247680 
Take  34567892000 


From  100000000 
Take     09999999 


Application. 

1.  What  was  the  age  of  a  man  in  the  year  1818,  who 
was  born  in  1777  ?  Ans.  41  years  old. 

2.  A  merchant  owes  5648  dollars,  and  pays  thereof  3460  ; 
how  much  is  yet  to  pay?  Ans.  2188  dolls. 

3.  D  having  on  hand  1260  barrels  of  flour,  sells  to  A 
320,  and  to  B  ^435  ;  how  many  barrels  are  yet  unsold  ? 

Ans.  505  barrels. 

4.  From   six   thousand  take  six  hundred,  and  tell  what 
remains.  Ans.  5400. 

5.  Suppose  a  boy  had  145  cents  given  him  at  one  time, 
15  at  another,  and  40  at  another  ;  and  he  gave  35  cents  for 
;i  penknife,  Mf)  Inr  a  slate,  (M  lor  paper,  and   30   for  apl  les  ; 
!io\v  many  rents  has  he  left?  Ans.   106  cents. 


MULTIPLICATION.  13 

SECTION  4. 

OF  MULTIPLICATION. 

MULTIPLICATION  is  either  simple  or  compound. 
Simple   multiplication  is  a  compendious  way  of  adding 
numbers  of  the  same  denomination  into  one  sum. 

The  number  to  be  multiplied,  is  called  the  multiplicand. 
The  number  multiplied  by,  is  called  the  multiplier. 
The  amount  produced,  is  called  the  product. 
The  multiplier  and  multiplicand  are  often  called  factors. 

MULTIPLICATION    TABLE. 


fl|  2]   3|  4|  5|  6|  7|  8| 

9|   10J   11|   12] 

2|  4|   6|   8|10J12|14|16| 

18|  20|   22    24 

3|   6|   9|12|15|18|21|24| 

27|   30|   33|   36 

4!  8|12!l6|20|24|28j32| 

36|   40|   44|  48 

5|10|15|20|25|30|35|40| 

45|  50|   55    60 

6|12|18|2430|36|42|48| 

54|  60|   66|  72 

7|14|21|2835|42|49|56| 

63|  70|  77|   84 

8|16|24|32|40|48|56|64j 

72|   80|   88|   96 

9|18|27|36|45|54|63|72| 

81|   90|   99|108 

10|20|30|40|50|60|70|80| 

90|100|110]120 

11|22|33|44|55|66|77|88| 

99|110|121|132 

12|*4|36|48|60|72|84|96|108|120|132144 

This  table  must  be  committed  to  memory,  with  great  care 
and  accuracy,  till  it  can  be  used  without  difficulty  or  hesita- 
tion by  the  scholar. 

Case  1. 
When  the  multiplier  does  not  exceed  12. 

RULE. 

Place  the  multiplier  under  the  multiplicand  ;  units  under 
units,  and  tens  under  tejis,  and  then  multiply  as  the  table 
directs,  taking  care  to  carry  1  for  every  10. 

EXAMPLES. 

46274963         24639576         3675432568 
24  S 


0-25*9920 


14  MULTIPLICATION. 

246H5761  4708-44753  964703024 

a  5  6 


74057343 


74020005       2901946808       246354276 
8  9  11 


Case  2. 

When  the  multiplier  exceeds  12. 
RULE. 


Multiply  by  each  figure  in  the  multiplier  separately,  be- 
ginning with  units,  taking  care  to  set  the  first  figure  in  each 
product  directly  under  its  own  multiplier.  Then  add  as  if. 
addition. 


EXAMPLES. 


2345601  68523047653 

234  2367 


9382404 
7036803 
4691202 

.,48870634 


PROOF. 


Method  1.  Change  the  multiplier  and  multiplicand ;  and 
then,  if  right,  the  product  from  this  multiplication  will  he 
equal  to  the  first. 

Method  2.  Cast  the  nines  out  of  each  factor  separately . 
set  dovvn  the  remainders  and  multiply  them  together  ;  casi 
rhe  nines  out  of  this  product,  and  note  the  remainder  ;  then 
;ast  the  nines  out  of  the  product,  and  if  right,  th*3  two  last 
remaindors  will  he  equal. 


MULTIPLICATION.  16 

EXAMPLES. 

Method  1.  Method  2. 

246         425        425 
425         246        246 


1230        2550       2550 
492        1700       1700 
984         850        850 


104550  104550  104550  6 

Note.    This  last  method  is  not  absolutely  certain ;  yet  the  probability 
is  so  great,  that  in  general  it  may  be  relied  on. 

3.  Multiply     5221          by          145          Ans.  757045 

4.  23430  230  5388900 

5.  3800920  80750  306924290000 
S.             89536925                      735  65809639875 
7.             78965987                     5893                  465346561391 

8.  What  will  75  bushels  of  wheat  come  to  at  1,15  cents 
per  bushel  ?  Ans.  86  dolls.  25  cents. 

9.  Bought  3950  Ibs.  of  coffee,  at  29  cts.  per  Ib.  what  must 
1  pay?  Ans.  1145  dolls.  50  cents. 

10.  There  are  12  pence  in  one  shilling.     How  many  are 
there  in  40  ?  Ans.  480  pence. 

Case  3. 

When  the  multiplier  is  the  exact  product  of  any  two  fac- 
tors in  the  multiplication  table. 

RULE. 

Multiply  the  given  sum  by  one  of  these ;  and  that  produci 
multiplied  by  the  other,  will  give  the  number  required. 

EXAMPLE. 

1.  Multiply  4236  by  16. 

41 

4X4=16. 


Product  67776 J 

2.  Multiply              871075     by     21  Ans.   18292575 

3.  2453642              36  88331112 

4.  43102              64  2758528 

5.  23645            144  3401880 
tt                                   12071               99 


48.00 
3600  by    400 
44000     550000 
663000      60000 

Ans.  4800 
1440000 
24200000000 
39780000000 

16  MULTIPLICATION. 

Case  4. 

When  there  are  ciphers  at  the  right  of  one  or  both  the 
factors. 

RULE. 

Omit  them  in  the  operation,  but  annex  them  to  the  product 

EXAMPLE. 

1.  Multiply  240  by  20.  24.0 

2.0 


2. 
3. 
4. 

Note.  When  the  multiplier  is  10,  the  product  will  be  found  by  add- 
ing one  cipher  to  the  multiplicand ;  if  100,  add  two  ciphers ;  if  1000 
add  three ;  &c. 

EXAMPLE. 

1.  Multiply          200     by  10  Ans.  2000 

2.  462  100  46200 

3.  879  1000  879000 

Application. 

1.  A  gentleman  owes  25  laborers  15  dollars  each  ;  how 
much  does  the  whole  come  to  ?  Ans.  375  dolls. 

2.  A  saddler  owes  his  journeyman  for  43  days'  work,  ai 
125  cents  per  day ;  how  much  does  he  owe  him  in  all  ? 

Ans.  53  dolls.  75  cts. 

3.  A  merchant  buys  440  yards  of  muslin  at  32  cents  per 
yard ;  how  much  does  the  whole  cost?  Ans.  140  dolls.  80  c. 

4.  A  farmer  sells  60  bushels  of  wheat  at  125  cents  per 
bushel ;  40  bushels  of  rye  at  85  cents  ;  34  of  corn  at  50 
cents  ;  how  much  is  he  to  receive  for  each,  and  how  much 
does  the  whole  amount  to  7 

Ans.  75,00  cents  for  the  wheat,  34,00  cents  for  the  rye 
17,00  cents  for  the  corn  ;  and  the.  whole  amounts  to  126,00 
<-<-nts,  or  126  dollars. 

.1.  A  dollar  is  equal  to  10  dimes,  and  a  ciime  is  equal  to 
10  cents  ;  how  many  dimes  and  cents  are  there  in  100  dol- 
Ans.  1000  dimes,  and  10,000  cents. 

6.  llo\v_  many  panes  of  glass  are  then:  in  a  house  that 
has  32  windows,  20  of  which  have  2 4  lights  <--ach,  and  the 
r<st  hav*'  J -:  ca<-h  .'  Ann.  096  panes. 


17 

7.  What  sum  is  equal  to  7525  multiplied  by  125? 

Ans.  940625. 

8.  A  has  250  dollars,  B  has  three  times  as  many,  and  C 
has  four  times  as  many  as  B ;  how  man)7  dollars  have  B  and 
C  each,  and  how  many  have  they  altogether  ? 

Ans.  B  has  750  dolls.  C.  3000  dolls,  altogether  $4000. 


SECTION  5. 

OF  DIVISION. 

Division  is  either  simple  or  compound. 
Simple  division  is  finding  how  often  one  number  is  con- 
tained in  another  of  the  same  name,  or  denomination.  * 
The  number  given  to  be  divided,  is  called  the  dividend. 
The  number  given  to  divide  by,  is  called  the  divisor. 
The  result,  or  answer,  is  called  the  quotient. 

Case  1. 
When  the  divisor  does  not  excised  12. 

RULE. 

Find  how  often  the  divisor  is  contained  in  the  first  figure 
or  figures  in  the  dividend,  under  which  set  the  result,  if  any 
remain,  conceive  it  as  so  many  tens  added  to  the  next  figure, 
and  then  proceed  in  the  samp  manner. 

Division  is  proved  by  multiplying  the  quotient  by^the  di- 
visor, and  adding  the  remainder,  if  any :  the  amount  will 
equal  the  dividend. 

EXAMPLES. 

Divisor  2)46578238    3)672245139     4)4756394344 


Quotient  23289119 
2 


Proof 


46578238 


224081713 
3 

672245139 


5)97036142     8)37846210 


12)64381259 


6)3824966     7)46825486     9)8297463813 


18 


DIVISION. 


Case  2. 
When  the  divisor  exceeds  12. 

RULE. 

Begin  with  as  many  of  the  first  figures  in  the  dividend  as 
will  contain  the  divisor.  Try  how  often  the  divisor  is  con- 
tained therein,  and  set  the  result  in  the  quotient. — Subtract 
the  product  of  the  divisor  multiplied  by  the  quotient  figure 
from  the  dividual  above,  to  this  remainder  annex  the  rvxr 
figure  in  the  dividend  for  a  new  dividual,  arid  so  proceed  tii: 
all  the  figures  in  the  dividend  are  brought  down. 

Note.  A  dividual  is  when  one  or  more  figures  of  the  dividend,  (in  the 
operation  of  long-  division)  are  divided  separately  from  the  rest. 


EXAMPLES. 


Divis.  42)9870       (235  Quot. 
84  42 


41)94979       (2316 
82  41 


147 
126 

210 
210 


470      . 
940 

9870  proof. 


123 


67 
41 


2316 
9264 

94956 
23  rem. 


269      94979  pr. 
246 

23  Rem, 


3.  Divide  29687624 
4.    47989536925 

Quotient. 

by  64  Ans.  463869 
735    65291886 

and  8  Rem. 
715 

5. 

4917968967 

2359 

2084768 

1255 

6. 

5374608 

671 

8009 

569 

7. 

19842712000 

175296 

113195 

81280 

M. 

5704392 

108 

52818 

43 

Case  3. 

When  the  divisor  is  the  exact  amount  of  any  two  factors 
in  the  table. 

RULE. 

Divide  the  given  sum  by  any  one  of  these,  and  the  quo- 
tient by  the  other. 


DIVISION. 
EXAMPLE. 

Divide  9870  by  42. 

6 ) 9870 

7 ) 1645 

235  Ans. 

Case  4. 
•Vhen  one  or  more  ciphers  stand  on  the  right  of  the  divisor. 

RULE. 

Omit  them  in  the  operation,  cutting  off  from  the  right  of 
the  dividend  as  many  figures,  taking  care  to  annex  them  to 
the  remainder. 

EXAMPLE. 

1 .  Divide  2564  by  200. 

2.00)25.64 


12    1  Rem. 
64 


Quot.    12  164  Rem. 

2.  Divide  87654  by  600  Ans.  146  54  Rem. 

3.  28347   '  80  354  27 

4.  137000       1600  85  1000 

Note.  When  the  divisor  is  10  the  quotient  will  be  had  by  cutting  off 
one  figure  from  the  right  of  the  dividend,  when  the  divisor  is  100  cut 
off  two  figures,  when  it  is  1000  cut  off  three  figures,  &c.  When  the 
figures  cut  off  from  the  right  of  the  dividend  are  digits,  they  are  to  be 
considered  as  so  much  of  a  remainder. 

EXAMPLE. 

I.  Divide  5640  by  10. 

1.0)564.0  Ans.  &64. 


L\  Divide  25654      byj    100  Ans.  256     54  Rem. 

'3.             876029             1000  876     29 

4.             800000           10000  80     — 

Application. 

1.  Several  boys  went  to  gather  nuts,  and  collected  4275  : 

when  they  had  divided   them,   each  had  855 ;  how  many 

boys  were  in  company?  Ans.  5. 


20  DIVISION. 

2.  If  2072  apple    trees  were  planted   in  28   rows,   how 
many  would  there  be  in  each  row  ?  Ans.  74. 

3.  If  45000  dollars  were  divided  among  75  persons,  how 
many  would  each  one  receive?  Ans.  600. 

4.  Into  how  many  parts  must  I  divide  the  number  8164. 
so  that  each  part  may  be  27,  leaving  the  remainder  10? 

'  Ans.  302. 

5.  There  is  a  certain  number,  to  the  double  of  which  if 
you  add  12,  then  5  times  that  sum  will  equal  150 ;  what  is 
that  number.  Ans.  9. 

6.  A  father  dying,  left  13440  dollars  to  be  divided  among 
his  6  sons  in  the  following  manner,  viz.  to  the  eldest  one- 
fourth  part,  to  the  second  one-fifth,  to  the  third  one-sixth,  to 
the  fourth  one-seventh,  to  the  fifth  one-eighth,  and  to  the 
youngest  the  remainder ;  what  was  each  son's  share  ? 

Ans.  1st  3360,  2d  2688,  3d  2240,  4th  1920, 
5th  1680,  6th  1552  dolls. 

7.  What   number^   if    multiplied   by  72084,  will   make 
5190048?  Ans.  72. 

8.  A,  B,  and  C,  engage  to  do  a  piece  of  work  for  228  dolls* 
which   together   they  accomplish  in  40  days  :   now  it  was 
previously  agreed  that  A  should  have  1 0  cents  per  day  more 
than  B,  and"  B  10  cents  more  than  C ;  what  was  each  man's 
share?  Ans.  A  80,  B  76,  C  72  dolls. 

9.  A  man  on  counting  his  money,  found  he  had  an  equal 
number  of  half  eagles,  (5  dollar  pieces)  half  dollars,  and 
quarter  dollars,  and  that  the  whole  amounted  to  1437  dol- 
lars 50  cents ;  how  many  pieces  of  each  kind  had  he  ? 

Ans.  250  of  each  kind. 

10.  The  crew  of  an  armed  ship,  consisting  of  the  cap- 
tain, mate,  and  40  men,  took  a  prize  worth  4550  dollars — 
now  it  was  agreed  that  the  captain  should  have  6  shares, 
the  mate  4,  and  each  seaman  1  share ;  what  did  each  one 
receive  ?  Ans.  The  capt.  546  dolls,  the  mate  364,  and 

each  seaman  91  dolls. 

As  but  few  examples  are  given  under  each  of  the  foregoing  rules,  it 
is  recommended  that  every  teacher  add  as  many  similar  ones,  as  may 
be  found  necessary  to  make  the  pupil  well  acquainted  with  their  appli 
cation,  and  both  expert  and  accurate  in  working  such  questions  as 
properly  belong  to  these  rules.  Every  experienced  teacher  is  well 
aware  that  until  this  knowledge  is  obtained  by  the  scholar,  every  at- 
tempt at  any  thing  farther  is  only  a  waste  of  time  and  money.  When 
this  knowledge  is  once  acquired,  the  future  progress  of  the  scholar  will 


FEDERAL    MONEY.  21 

oe  pleasant  and  rapid.  The  teacher  will  then  be  justly  rewarded  lor  his 
labor  and  trouble  in  this  part,  by  the  approbation  of  parents,  and  the 
gratitude  of  his  scholars,  who  will  have  acquired  the  necessary  qualifi- 
cations (accuracy  and  expertness)  for  the  great  variety  of  studies  and 
avocations  in  future  life,  which  require  the  aid  of  arithmetic  and  math- 
ernatics. 


PART  II. 
ARITHMETIC  IN  MIXED  OR  COMPOUND  NUMBERS. 


SECTION  I. 

FEDERAL  MONEY 

Is  so  called  from  its  being  the  general  currency  establish- 
ed by  the  Federal,  or  United  States'  government,  and  is  justly 
considered  superior  to  every  other  kind  of  currency  now  in 
use  for  its  simplicity  and  plainness. 

Standard  weight  as  establieh- 
ed  by  law. 

Its  denominations  are,  dwt.      gr. 

10  Mills  make  1  Cent 

10  Cents  1  Dime      1  16 f,  >  gn 

•  10  Dimes  or  100  Cents         1  Dollar  17  '  l|    < 
10  Dollars                               1  Eagle  11          4|    5  n  }, 

i  Eagle     5  14J-    $  b 

From  this  table  it  will  readily  be  seen,  that  addition,  sub- 
traction, multiplication,  and  division  of  federal  money  may 
be  performed  as  if  they  were  whole  numbers.  It  will  also  be 
seen  that  to  reduce  any  number  of  mills  to  cents,  it  is  only 
necessary  to  point,  or  cut  off  the  last  figure,  as  100  mills 
=  10,0  cents;  an^l  cents  in  the  same  way  to  dimes,  as  100 
eents=:10,0  dimes,  and  dimes  to  dollars,  as  100  dimes=:10,0 
dollars,  and  dollars  to  eagles,  as  100  dollars— .10,0  eagles; 
and  also  that  eagles  may  be  brought  to  dolls,  and  dolls,  to 
dimes,  &c.  by  adding  a  cipher  to  each  one,  as  10  E.=  100 
dolls.  =  1  OOOd.  =  1  OOOOc.— 1  OOOOUm. 

In  all  calculations  in  federal  money,  according  to  com- 
mon custom,  it  is  usual  to  omit  the  names  of  eagles,  dimes, 


22  FEDERAL  MONEY. 

and  mills,  and  only  to  reckon  by  dollars  and  cents;  the 
eagles  being  considered  as  so  many  10  dollars,  the  dimes  as 
so  many  10  cents,  and  the  mills  as  fractional  parts  of  the 
cent.  See  the  following 

.     TABLE. 


i    s 


Si       Dolls. 

1,  2  5  1,25  cents 

3  4,  1  2  i  34,12  and  a  fourth  cents 

4  5  6,  2  5  i          456,25  and  a  half  cents 
8  2  6  4,  7  5  |        8264,75  and  three-fourth  cents 
Note.    1.  In  addition,  subtraction,   multiplication,   and  division  of 
federal  money,  if  the  sums  are  dollars  only,  the  amount,  remainder, 
product,  or  quotient,  will  be  dollars;  but  when  the   sums  consist  of 
dollars  and  cents,  or  cents  only,  the  two  first  right  hand   figures  arc 
cents,  and  all  the  rest  are  dollars. 

2.  When  fractions  of  cents  are  used  according  to  the  above  table, 
every  four  of  them  make  one  cent :  in  adding,  or  subtracting  these, 
we  carry  one  for  every  four ;  and  in  multiplying,  the  upper  figure,  called 
the  numerator,  is  to  be  multiplied  by- the  multiplier,  and  divided  by  the 
lower  figure,  called  the  denominator. 

EXAMPLES    OF    ADDITION. 

Edd cm  DC  DC 

25,6,4,8,2  5675,25  53258,75* 

24.7.6.2.4  2386,63  93620,33^ 

63.8.1.3.5  3972,80  30176,56$ 

92.2.3.4.6  7285,75  .  27532,35 


206,4,5,8,7 


SUBTRACTION. 

E  d  d  c  m  DC  D 

83,6,5,3,5 
32,9,3,7,5 


50,7,1,0,0 


FEDERAL    MONEY.  23 

MULTIPLICATION. 

Edd cm  DC  DC 

23,6,3,5,7  2637,25  6378,75£ 

369 


70,9,0,7,1 

DIVISION. 

Edd  cm  DC  DC 

2)63,3,8,6,2  5)3632,75  8)82750,33 


31,6,9,3,1 


Promiscuous  Questions. 

1.  Add  25  eagles,  62  dollars,  8  dimes,  75  cents,  and  5 
mills.  Ans.  3.13d  55c  5m. 

2.  A  person  deposited  at  bank  1055  dollars  in  notes,  260 
dollars  in  gold,  3650  dollars  in  silver,  and  2,50  cents :  how 
much  is  the  amount?  Ans.  4967d  50c. 

3.  Bought  a  barrel  of  sugar  for  39  dollars  87 £  cents,  a 
bag  of  coffee  for  22  dollars  18J  cents,  and  a  pound  of  tea 
for  2  dollars  12^  cents ;  how  much  do  they  all  cost? 

Ans.  64d  18jc. 

4.  Bought  goods  to  the  amount  of  645  dollars  95 1  cents, 
and  paid  at  the  time  of  purchase  350  dollars ;  how  much  re- 
mains to  be  paid  ?  Ans.  295d  95 f. 

5.  A  man  lent  his  friend  1000  dollars,  and  received  at 
sundry  payments,  first  160  dollars  25  cents,   second  285 
dollars  66-J  cents,  third  300  dollars  28|  cents ;  what  remains 
yet  to  be  paid?  Ans.  253d  79fc. 

6.  What  is  the  product  of  102  dollars  19  cents,  multiplied 
by  120?  Ans.  12262d  80c. 

7.  What  will  16  barrels  of  flour  amount  to,  at  4  dollars 
50  cents  per  barrel  ?  Ans.  72d. 

8.  How  much  will  132  pieces  of  calico  come  to,  at  11 
wllars  37  i  cents  a  piece?  Ans.  2293d  50c. 

9.  What  is  the  quotient  of  6022  dollars  50  cents,  divided 
by  5?  Ans.  1204d  50c. 

10.  A  butcher  bought  18  beef  cattle  for  252  dollars  90 
cents;  how  much  did  he  pay  for  each?          Ans.  14d  05c. 

11.  Bought  45  yards  of  linen  for  22  dollars  50  cents, 
what  was  the  price  of  one  yard?  Ans.  50cU- 


24  COMPOUND    ADDITION. 

12.  If  25  men  expend  15555  dollars  50  cents  in  the  erec- 
tion of  a  bridge,  how  much  has  each  one  to  pay,  if  the 
shares  are  equal  ?  Ans.  622d  22o 

Having  treated  of  federal  money  separately,  inasmuch  as  it  requires 
to  be  well  understood,  seeing  it  is  the  general  currency  in  the  United 
States;  we  now  proceed  to  the  other  parts  of  mixed  numbers,  or  as  they 
are  frequently  termed,  divers  denominations. 


SECTION  2. 

OP  COMPOUND  ADDITION. 

COMPOUND  Addition  is  the  collecting  together,  and  thereby 
ascertaining  the  amount  of  several  quantities,  of  divers  de- 
nominations. 

RULE. 

Place  the  numbers  in  such  a  manner,  that  all  of  the  same 
denomination  may  stand  directly  under  each  other,  then  be- 
ginning with  the  lowest  denomination,  add  as  in  whole  num- 
bers, carry  at  that  number  which  will  make  one  of  the  next 
greater;  set  down  the  remainder  (if  any)  and  so  proceed  till 
nil  are  added. 

PTCOOF  as  in  simple  addition. 

ENGLISH  MONEY. 

The  denominations  are,  pounds,  shillings,  pence  and  far- 
things, and  are 
Thus  valued : 
4  farthings  (marked  qr.)  make      1  penny  (marked)     d. 

12  pence 1  shilling  *. 

20  shillings 1  pound  £. 

TABLE    OF    SHILLINGS. 

s.  £.  s. 

20  shillings  make  1  00 

30  ....  1  10 

40  ....  2  00 

50  ....  2  10 

60  ....  3  00 

70  ....  3  10 

80  ....  4  00 

90  .           .     .  4  10 

100  .  5  00 


PENCE    TABLE. 


d. 

20  pence  make 

30  .     .     . 

40  ... 

50  ... 

no  .    .    . 

70  ... 

90  ... 

00  ... 

100  ... 


*.  d. 

1  8 

2  6 

3  4 

4  2 

5  0 

5  10 

6  8 

7  6 
a  4 


COMPOUN  D  A  DDITIO3N .  25 

EXAMPLE* 

£.   s.  d.     £.   s.  d.  qr.  £.   s.  d.  qr. 

12-*6  11  8    35678  11   9  \  2368  17  5  4 

9462  8  4    37562  18  7  $  3969  19  11  | 

3215  10  6    63497  15  10  i  9386  14  6  $ 


£.13934  10     6 


raor  WEIGHT. 

This  weight  is  used  for  jewels,  gold,  silver,  and  liquors. 

The  denominations  are,  pounds,  ounces,  pennyweights,  and 
grains. 

Thus  valued. 

24  grains  (gr.)  make      .      1  pennyweight  dwt. 

20  pennyweights      .       .      1  ounce  oz. 

12  ounces       ....      1  pound  ft. 

EXAMPLES* 

ib.oz.dwt.gr.  lb.  oz.  dwt.  gr.  lb.oz.dwt.gr. 

4  10  15  16  5  8  11  16  45  17  11 

8  6  10  11  9  10  15  21  9  6  12   9 

6  9  14  23  6  11  18  17  18  11  19  23 


20     3     1     2 


AVOIRDUPOIS  WEIGHT 

This  weight  is  used  for  heavy  articles  generally,  and  all 
metals  but  gold  and  silver. 

Tlie  denominations  are,  tons,  hundreds,  quarters,  pounds 
1  ounces,  and  drams. 

Thus  valued. 

16  drams  (dr.)  make     ...  1  ounce  oz. 

16  ounces        1  pound  lb. 

28  pounds 1  quarter  qr. 

4  quarters,  (or  112  lb.)    .     .  1  hundred  cwt. 

20  hundreds         1  ton  T. 

C 


2f>  COMPOUND  ADDITION 

EXAMPLES. 

T.  cwt.  qr.  Ih.  oz.  dr.        T.  cwt.  "  qr.  Ib.  oz.  dr. 

10  16  2  24  9  12      856  12  3  19  11  10 

15  11  1  15  12  9      537  19  1  23  8  9 

85  8  3  19  13  13      638  10  2  21  12  6 

18  15  1  14  10  8      897  19  3  27  15  15 


APOTHECARIES'  WEIGHT. 

This  weight  is  used  by  apothecaries  in  mixing  medicines, 
.' "t  they  buy  and  sell  by  avoirdupois. 

The  denominations  are,  pounds,   ounces,  drams,  scruples, 
and  grains. 
Thus  valued. 

20  grains  (gr.)  make     ...     1  scruple  sc.    or    9 

3  scruples 1  dram  dr.          3 

8  drams         ,   .     1  ounce  oz.           3 

12  ounces 1  pound  Ib.          ft 

EXAMPLES. 

Ib.  oz.  dr.  sc.  gr.        Ib.  oz.  dr.  sc.  gr. 

*  5  2  1   16        17  5  7  2  14 

3  11   7  2  19        80  3  2  1  16 

6  8  6  1   12        85  10  3  2  5 

5  2  4  2   9        36  6  2  1  15 


CLOTH  MEASURE. 

By  this  measure,  cloths,  ribands,  &c.  are  measured. 
The  denominations  are,  Ells  French,  Ells  English,  Elh 
Flemish,  yards,  quarters,  and  nails. 

Thus  valued. 
4  nails  (na.)  make     ...      1  quarter  qr. 

4  quarters        I  yard  yd. 

3  quarters        i  Kll  Flemish        E.  FL 

5  quarters         1   Mil  Kn^iish         E.  En. 

0  quarters        i   Kll  French         E.  Fr. 

EXAMPLES. 

Yd.     qr.    na.          E.  1?l.  qr.    nn.          E.  Fr.  qr.  na.  E.  En.  qr.  na 

56     2     2            HO     2     3            16     4  2  53     4     3 

sG      13            18      1      2             17     5  1  53     3     ? 

33     3     2            36     2.1             80     2  2  32     2      1 

1*8     2      1             :W     21             13     3  3  81      0     0 


COMPOUND  ADDITION.  27 

LONG  MEASURE. 

By  this,  lengths  and  distances  are  measured. 

The  denominations  are,  degrees,  leagues,  miles,  furlongs, 
poles,  rods  or  perches,  yards,  feet,  inches,  and  barley- 
corns. *y 

Thus  estimated. 

3  barleycorns  (be.)  make  1  inch  in. 

12  inches 1  foot  ft- 

3  feet 1  yard  yd. 

5£  yards  (or  16J  feet)  .  1  rod,  pole,  or  perch  P. 

40  poles  rods,  or  perches,  >  fo  ,  /wn 
(or  220  yards)  $ 

8  furlongs  (or  320  poles,  ?  -,      -,  a/r 

?-r^n     i    \  t  1  mile  •*«• 

or  17 GO  yds.)  $ 

3  miles 1  league  L. 

60  geographic,  or  )     .,  , 

flft-f  *  *  £  miles         1  degree  deg. 

69i  statute  ^ 

360  degrees  make  a  circle,  or   the  circumference  of   the 

earth. 
A  hand  is  a  measure  of  4  inches,  and  a  fathom  of  6  feel. 

EXA3fPLES. 

deg.  m.  fur.  po.   yds.  ft.    in.    be.  L.'   M.  fur.  yds.  ft.  in. 

50  30     5  15     2     2     9     2  5     2     6     75  2  11 

60  25     7  12     4     1   10     1  3     1     4     95  1  9 

75  35     2  92281  2     1      3     15  2  8 

20  55     6  8     1      1   11     2  125  200  1  6 


LAND  MEASURE. 

By  this  the  quantity  of  land  is  estimated. 

The  denominations  are,  acres,  roods,  perches,  yards,  anl 
feet. 

Thus  rated. 

9  feet  (ft.)  make     .     .     .     .     .     1  yard  yd. 

30^  yards          ..._....!  perch  P. 

40  perches 1  rood  R. 

4  roods  (or  160  perches)        .     .     1  acre  A. 


28  COMPOUND  ADDITION. 

EXAMPLES. 

A.  R.  P.  A.  R.  P.  A.  R.  P. 

25  3  20  265  2  15  246  3  29 

33  1  16  375  1  29  762  1  12 

33  2  34  860  3  39  632  2  11 

68  1  39  632  2  20,  357  3  20 


CUBIC,  OR  SOLID  MEASURE. 

By  this,  wood  and  other  solid  bodies  are  estimated. 
The  denominations  are,  cords,  tons,  yards,  feet,  and  inches 

Thus  rated : 

1728  inches  (in.)  make     ...     1  foot  ft. 

27  feet       .......     1  yard  yd. 

40  feet  of  round  timber,  or  >        ^  tQQ  y 

50  feet  of  hewn  timber         $ 
128  feet 1  cord  cor. 

EXAMPLES. 

Co.  ft.    in.  T.  ft.  in.  T.  ft.  in. 

4  112  1260  6  39  1384  23  *12  1400 
6  84  1500  2  26   526  68  45  1600 
8  127  1700  8  18   260  82  49  1700 

5  63   403      3  12  1100      96  18    50 


TIME. 

This  relates  to  duration. 

The  denominations  arc,  years,  months,  iceeks,  days,  hours, 
minutes,  and  seconds. 

The  relative  differences  are  these. 

60  seconds  (sec.)  make       ...  1  minute  mi. 

(>0  minutes 1  hour  h. 

24  hours 1  day  d. 

7  d.'iys 1  week  w. 

4  weeks 1  month  M. 

12  months,  5'^  weeks,  or  365  >    .  .  vr 
days  and  6  hours             \ 

Note.    The  solar  year,  according  to  the  most  °.xnrt  observation,  con 
tains  3G5dav.s,  5  JuMirs,  4ft  minutes,  57  seconds. 


COMPOUND  ADDITION  21) 

TTie  number  of  days  in  the  12  calendar  months,  is  thus  found: 

Thirty  days  are  in  September, 
April,  June,  and  November  ; 
February  hath  twenty-eight  alone, 
And  all  the  rest  have  thirty-one. 

JNote.     Every  fourth  year  is  called  bissextile,  or  leap-year,  in  which 
February  has  twenty-nine  <days. 

EXAMPLES. 

Y.  M.  da.  h.  mi.  sec.  Y.  da,,  h.  mi.  sec. 

22  10  25  16  34  55  4  350  15  19  5 

34  6  16  20  48  33  2  268  13  54  38 

46  9  13  23  59  59  6  350  22  50  50 


MOTION. 

This  relates  to  the  measure  of  circles. 

The  denominations  are,  circles,  (or  revolutions)  signs,  de- 
grees, minutes,  and,  seconds. 
The  relative  differences  are, 

60  seconds  (sec.)  make       -       1   minute  mi.  or  ' 

60  minutes      .....        1   degree  deg.  or  ° 

30  degrees       -       -       -  .    -       1   sign  sig. 

12  signs  (or  360  degrees)          1   circle 

EXAMPLES. 

sig.  deg.  mi.  sec.  sig.  °  '  " 

2  24     48     58  3  20  30  40 

2  29     59     59  2  25  35  45 

3  21     20     20  3  2d  38  58 


LIQUID  MEASURE. 

This  is  used  for  measuring  wine,  spirits,  cider,  beer,*&c. 
The  denominations  are,  tuns,  pipes  or  butts,  hogsheads,  bar- 
rels, gallons,  quarts,  pints,  and  gills. 

Thus  estimated. 

4  gills  (gi.)  make       -       -       1  pint  pt. 

2  pints         -        -       -       -       1  quart  qt. 

4  quarts  1   gallon  gal. 

63  gallons     -        -       -       -       1   hogshead  hhd. 

2  hogsheads  1   pipe,  or  butt  pi.  bt 

2  pipes  (or  four  hogsheads)      1   tun  T. 

Note.     By  a  law  of  Pennsylvania,  32^  gallons  make  a  barrel,  and 
16  gallons  make  a  half  barrel. 

C2 


30 


COMPOUND  SUBTRACTION. 
EXAMPLES. 


T.hhd.gal.  qt.  pt. 

4     3     53     2  I 

6     2     25     3  1 

8     1      62     1  1 


T.  hhd.  gal. 
24     2     33 
19     3     54 
34     1      50 


DRY  MEASURE. 

This  is  used  for  measuring  grain,  salt,  fruit,  &c. 
Tht  denominations  are,  bushels,  pecks,  quarts,  and  pints. 

Thus  estimated. 

2  pints  (pt.)  make         -          -          1   quart  qt. 

8  quarts        -         -         •-  1   peck  P. 

4  pecks  (or  32  quarts)  1  bushel  bu. 


bu.  P.  qt. 

25  2     4 

36  3     6 

34  1      2 

78  2     7 


EXAMPLES. 

bu.  P.  qt. 

256  3  6 

243  1  6 

468  3  1 

584  2  7 


bu.  P.   qt. 

34156  3      7 

2003  1      2 

9£0  3     6 

4809  0     0 


SECTION  3. 
OF  COMPOUND  SUBTRACTION. 

COMPOUND  SUBTRACTION  is  the  taking  a  lesser  numher 
from  a  greater,  of  divers  denominations,  and  thereby  finding 
l he  difference. 

RULE. 

Set  down  the  lesser  number  under  the  greater,  as  in  com- 
pound addition.  Then  beginning  with  the  lowest  number, 
subtract  as  in  subtraction  of  whole  numbers :  when  the 
lower  number  is  greater  than  the  upper,  take  it  from  as 
many  of  that  denomination  as  will  make  one  of  the  greater, 
and  to  the  remainder  add  the  upper  number,  set  down  the 
amount  and  carry  one  to  the  next,  and  so  proceed  till  all  are 
subtracted. 


COMPOLLN  1>  SUBTRACTION 

PROOF. 
Add  the  remainder  to  the  lower  line. 

EXAMPLES. 

£.  s.  d-. 
From  25(T  15  6 
Take  129  12  8  $ 


31 


T.  cwt.  qr.  Ib.  oz.  dr. 
From  246  15  2  18  11  5 
Take  89  16  1  24  8  15 


Rem.  127     2     9  } 


Proof  256   15     6 


mi.  fur.  P.   ft.    in.  be. 
From  250    4    24    10      61 
Take   125    6    30      5    10    2 


D.  h.  mi.  sec. 
From  325  18  30  24 
Take  236  20  45  50 


bn.  ,  P.  qt.  pt. 
From  204  2  6  1 
Take  150  3  2  0 


sig.  deg.  mi.  sec. 
From  6  16  32  29 
Take  3  . 24  16  48 


T.  hhd.  gal.  qt.  pt. 
From  50  2  45  2  1 
Take  20  3  60  3  0 


A.  R.  P. 
From  1658  2  Vj 
Take  1249  3  ?4 


Promiscuous  Questions  in  Compound  Addition  and 
Subtraction. 

1.  A  merchant  bought  five  pieces  of  linen,  containing  as 
follows:  No.  1,   36  yards  3  quarters  2  nails;   No.  2,   45 
yards  1  quarter  3  nails  ;  No.  3,  48  yards  2  quarters  1  nail ; 
No.  4,  52  yards  3  nails  ;  No.  5,  64  yards  2  quarters  ;  how 
many  yards  were  in  all  ?  Ans..  247 yds.  2qr.  Ina. 

2.  Sold  5  head  of  beef  cattle,  at  the  following  prices,  viz : 
the  first  far  67.  2s.  4d.  the  second  for  5Z.  10s.  9%d.  /he  third 
for  11.  the  fourth  for  SI.  10s.  6d.  the  5th  for  Ql.  2s.  6d.  and 
received.  2.21.  10s.  6d.  in  ready  payment,  and  a  note  for  the 
remainder  ;  how  much  did  the  cattle  cost,  and  for  how  much 
was  the  note  given  ? 

Ans.  The  cattle  cost  367.  6s.  l^d.  and  the  note  wastfor 
13/.  15*.  l%d. 


32  COMPOUND  MULTIPLICATION. 

3.  A  silversmith  bought  26lb.  9oz.  lOdwt.  of  silver,  and 
.  wrought  up  ISlb.  \6dwt.  Wgr.  how  much  has  he  left  ? 

Ans.  Sib.  Soz.  ISdwt.  llgr. 

4.  A  physician  bought  Qlb.  Woz.  6dr.  2sc.  (apothecaries' 
weight)  of  medicine,  and  has  used  4Z&.  Soz.  4dr.  Isc.  I7gr. 
what  quantity  has  he  yet  remaining  ? 

Ans.  2lu.  5oz.  2dr.  Osc.  3gr. 

5.  William  was  born  on  the  15th  day  of  January,  1816, 
at  6  o'clock  in  the  morning,  and  Charles  was  born  on  the 
20th  of  March,  1817,  at  9  in  the  evening  ;  how  much  older 
.6  William  than  Charles  1  Ans.  1  year,  2mo.  5d.  15h. 

ft  An  innkeeper  bought  four  loads  of  hay,  weighing  as 
lollowing,  viz.  first  load,  19  hundred  2  quarters  and  14  Ib. 
second  load,  16  hundred  3  quarters  18  Ib.  third  load,  22 
hundred  and  24  Ib.  fourth  load,  24  hundred  and  1  quarter  • 
how  much  hay  in  all  ?  Ans.  4  tons  2  hundred. 

7.  From  a  piece  of  broadcloth  which  at  first  measured 
55  yds.  I  sold  to  A  5J  yds.  to  B  6|,  to  C  7|,  to  D  a  quan 
tity  not  recollected,  and  to  E  just  half  as  much  as  to  D  ;  on 
Pleasuring  the  remainder,  I  found  there  was  20J  yds.  left ; 
how  many  yards  did  D  and  E  each  receive  1 

Ans.  D  10,  and  E  5  yds. 

8.  A  wine  merchant  bought  1   pipe  2  hhds.  and  3  qr. 
casks  of  wine,  each  26  galls.  ;  of  these  he  sold  1  hhd.  and 

-jr.  casks;  he  also  found  that  the  pipe  had  leaked  17  galls, 
the  remaining  hogshead  11,  and  the  cask  5-J  ;  how  many 
gallons  did  he  buy,  and  how  many  had  he  left  ? 

Ans.  Bought  330  galls,  left  18l£  galls. 

9.  Bought  4  pieces  of  cloth,  the  two  first  measured  9  Ells 
Fr.  3  qr.  2  na.  each,  and  the  two  last  8  Ells  Fr.  2  qr.  3  na. 
each,  of  these  I  sold  40 5  yards  ;  how  much  have  I  left  ? 

Ans.  13y.  2  qr.  2  na. 


SECTION  4. 
OF  COMPOUND  MULTIPLICATION. 

COMPOUND  MULTIPLICATION  teaches  to  multiply  any  given 
quantities  or  numbers  of  divers  denominations. 

Case  1. 
When  the  multiplier  does  not  excer<i  12. 


COMPOUND  MULTIPLICATION 
RULE. 


33 


Begin  by  multiplying  the  lowest  number  first,  as  in  inte- 
gers ;  divide  the  product  by  that  number  which  will  make 
one  of  the  greater.  Set  down  the  remainder,  if  any,  and 
carry  the  quotient  to  the  next  number. 


KXAMPLES. 


Ib. 

14 

£.  5.  d. 
24  10  6 

cjr.        T.  cwt.  qr.  Ib.  oz.  dr. 
\        48  14  1   4  12  11 
2                      3 

ft. 
1 
11 

2  )  49  1   1 

24  10  6 

4  Proof. 

oz.  dwt.  gr. 
4   11   11 
5 

bu.  pc.  qt.     hhd.  gal.  qt. 
24  3  7       25  48  3 

8 

* 

8 

mi.  fur.  p. 
24  6  34 

8 

yds.  y/.  in.  be.      A.  R. 

24  282       89  3 
6 

P. 

26 
9 

to/. 

48 

y?c.  #. 
3  6 
11 

d.   h.  mi.  sec.      Y.  _m.  w*  d. 
84  19  38   15      125  8  3  4 
9               12 

Application. 


I.  5  yards  of  cloth  at 

.2.  9  '     do.  at 

3.  11  bushels  of  flax-seed  at 

4,  12      do.      clover-seed  at 


£.    s.d. 
264 

1  2   \>\ 
12   9j 

2  4  2| 


Ans. 


£. 
11 

10 

7 


s.  d. 

11   8 
2,  84 
0  8i 


26   10  6 


Case  2. 

When  the  multiplier  exceeds  12,  uut  is  the  exact  produd 
of  any  two  factors  in  the  table-1. 


34  COMPOUND  MULTIPLICATION. 

RULE. 

Multiply  the  given  sum  by  any  one  of  these,  in  the  same 
•nanner  as  above,  and  the  product  by  the  oilier. 

EXAMPLES. 

£.     s.    d.  qr. 

Multiply    12     8     6  £  by  18=3x6 
3 


37     5     6  | 
6 

223  13     4  £ 
Application. 

1.  Multiply  4T.  Scwt.  Iqr.  16ZZ>.  Soz.  Wdr.  by  36. 

Arts.   150  T.  2cwt.  Iqr.  lib.  6oz.  8dr. 

2.  120Z.  (5s.  9d.  by  24.       A/is.  238SZ.  2s.  Od. 

3.  24  T.  4e-7/tf.  2<p%  lib.  by  48. 

Aws.   1162T.  Wcwt.  Oqr.  OZ/;. 

Case  3. 

When  the  multiplier  is  not  the  exact  product  of  any  two 
factors. 

RULE. 

Multiply  as  in  the  last  case,  by  any  two  factors  that  will 
come  the  nearest  to  the  multiplier,  but  less  ;  and  add  for  the 
deficiency. 

EXAMPLES. 
bll.     pC.  qt. 

Multiply    12     2     4  by  17 

4x4+1  =  17 


50     2     0 
4 

202     0     0  =  16 
12     2     4=    1 


214     ^     4     17 


3.    Multiply  H/>.  4/>.  I2?ni.  f>«v.  by  2JK 

AH*.   237  />.   I/?.  507/w. 


COMPOUND  DIVISION.  35 

Case  4. 

When  the  multiplier  exceeds  the  product  of  any  two  fac- 
tors in  the  table. 

RULE. 

Multiply  by  the  units  figure,  as  in  case  1,  and  set  down 
the  amount;  again  multiply  the  given  sum  b}*  the  figure  of 
tens,  and  that  product  by  10,  and  place  tins  amount  under 
the  first ;  again  multiply  by  the  figure  of  hundreds,  and  the 
product  by  10  and  10,  which  set  down  under  the  other  pro- 
ducts-— in  the  same  way  for  thousands,  by  three  tens,  &c. 

EXAMPLES.  - 

s.    d.  2626 

Multiply    2     6  by  245                                        4                 2 
5  

10  0  50 

12     6  first,  or  units  product  10  10 

500  second,  or  tens  do. 

25     0     0  third,  or  hundreds  do.      500      2100 

10 

Ans.  30  12     6  total.  

25  0  0 

£.  s.     d.  £.      s.    d. 

2.  Multiply          14     6  by  240  Ans.  174     0     0 

3.  123        117  130     3     3 

4.  126        275  309     7      6 


SECTION  5. 

OF  COMPOUND  DIVISION. 

COMPOUND  DIVISION,  teaches  to  divide  any  sum  or  quan- 
tity of  divers  denominations. 

Case  1. 
When  the  divisor  does  not  exceed  12. 

RULE. 

Divide  the  highest,  or  left-hand  denomination;  if  any  re- 
mains, multiply  by  that  number  which  will  reduce  it  to  the 
next  highest,  add  this  product  to  the  second,  then  divide  as 
Df.'ibre,  and  so  proceed  till  all  are  divided. 


30  COMPOUND  DIVISION. 

PROOF — By  compound  multiplication. 

EXAMPLES. 

£.        s.     d.  qr.  £.       s.     d.  qr 

2)465     10     6  i  3)563     15     4     £ 


Quotient  232     15     3  £ 
2 


Proof       465     10     6  £ 

T.    cwtf.  f/r.   Ib.  yds.       ft.       in. 

6)91     16     1     14  5)960          1          9 


T.  Mid.  gal.  qt.        .  w.    d.     h.     mi.    sec. 

8)468     1     48     3  10)30     6     18     48     50 

Case  2. 

When  the  divisor  is  the  exact  product  of  any  two  factors 
in  the  table. 

RULE. 

Divide   first   by  one  as  above,  and  the  quotient  by  the 
other. 

EXAMPLES. 

6)224  12     6  by  30=6x5 


5)   37     8     9 


Quotient 

7     9 

9 

£. 

s. 

d. 

£. 

s. 

d. 

2.  Divide 

134 

18 

S 

by  44 

Ans.  3 

1 

4 

3. 

9H4 

0 

0 

144 

6 

16 

8 

4. 

474 

0 

0 

72 

6 

11 

8 

Case  3. 

When  the  divisor  is  not  the  exact  product  of  any  two  (ac- 
tors in  the  table,  or  eA,:ee.ds  them. 

RULE. 

Divide  the  highest  denomination  in  the  ^iv<-n  sum,  in  the 
.s.-iuic  manner  as  in  case  2,  of  whole  numbers;  reduce  I  lit 
remainder,  if  any,  to  the  next  lower  denomination,  adding  it 


COMPOUND  DIVISION.  37 

to  the  number  of  the  same  denomination  in  the  given  sum  ;  di- 
vide this  in  the  same  manner,  and  so  proceed  till  all  are  divided. 

EXAMPLES. 

1.  Divide  264Z.  10s.  7$d.  by  25. 

25)264/.  10s.  l^d.  (Wl.lls.l^d.  Ans. 
25 


20 

25)290(11 
25 

~40 
25 

15 
12 

25  )  187  (  7 
175 

Is 

4 

25)50(2 

50 
£.      s.     d.  £.    s.     d. 

2.  Divide     409  13  9  by  345  Ans.  139 

3.  232     4  9        524  8  l(5i 

4.  3236  12  4J      654  4  l3  llj 

5.  132     0  8  68  1   18  10 
Promiscuous  Questions,  for  exercise,  in  the  foregoing  rules 

of  Compound  Addition,  Subtraction,  Multiplication  and 
Division. 

1.  What  is  the  value  of  672  yards  of  linen  at  2s.  5d.  per 
yard?  Ans.  81Z.  45. 

2.  A  goldsmith  bought  11  ingots  of  silver,  each  of  which 
weighed   4db.  loz.    I5dwt.  22gr.    how  much   do   they   all 
weigh?  Ans.  45Z&.  loz.  15dwt.  2gr. 

3.  Bought  8  loads  of  hay,  each  weighing  1  ton  2  hundred 
3  quarters  16  pounds;  how  much  hay  in  all? 

Ans.  9  ton  3  hun.  161b. 

4.  Divide  9  ton  3  hundred  161b.  into  8  shares. 

Ans.  1  ton  2  hun.  3qr.  16lb. 

5.  Bought  15  tracts  of  land,  each  containing  300  acres 
2  roods  and  20  perches  ;  what  is  the  amount  of  the  whole  ? 

Ans.  4509  acres  1  qr.  20  rods. 
D 


38  COMPOUND  DIVISION. 

6.  Divide  a  tract  of  land  containing  4509  acres  1   rood 
and  20  perches  equally  among  15  persons;  what  is  each 
one's  share?  Ans.  300  acres  2  roods  20  perches. 

7.  Bought  179  bushels  of  wheat  for  201  dollars  37^  cts. 
what  is  it  per  bushel?  Ans.  I  doll.  12^  cents. 

8.  If  a  man  spends  7  pence  per  day,   how  much  will  it 
amount  to  in  a  year?  Ans.  10Z.  12s.  lid. 

9.  What  is  the  value  of  1000  bushels  of  coal  at  10|  cents 
per  bushel?  Ans.  105  dolls. 

10.  Bought  135  gallons  of  brandy  at  1  dollar  arid  62£ 
cents  per  gallon,  which  was  sold  for  2  dollars  and  5  cents 
per  gallon ;  required  the  prime  cost,  what  it  was  sold  for,  and 
the  gain?  Ans.  prime  cost  219  dolls.  37^  cts. 

sold  for  276  dolls.  75  cts.  gain  57  dolls.  37 £  cts. 

11.  If  27  cwt.  of  sugar  cost  47/.  12s.  W$d.  what  cost  1 
cwt.  ?  *  Ans.  ll.  15s.  3^d. 

12.  Suppose  a  man  has  an  estate  of  9708  dollars,  which 
he  divides  among  his  four  sons  :  to  the  eldest  he  gives  f ,  and 
to  the  other  three  an  equal  share  of  the  remainder  ;  what  is 
the  share  of  each? 

Ans.  eldest  son,   3883  dolls.  20  cents,  other  sons,  each 
1941  dollars  60  cents. 

13.  A  dollar  weighs   lldwt.  Sgr.  what   will   45  dollars 
weigh  at  that  rate?  ^Ans.  39oz. 

14.  An  eagle  of  American  gold  coin  should  weigh  lldwt. 
tigr. — now  150  were  found  to  weigh  84oz.  7dwt.  20gr.  how 
much  was  this  over  or  under  the  just  weight  ?   Ans.  &gr.  over. 

15.  What  cost  2^-  cwt.  of  sugar,  at  13  cents  3  mills  per 
pound  ?  Ans.  37  dolls.  24  cts. 

16.  A  merchant  deposited  in  bank  35  twenty-dollar  notes, 
63  eagles,   284   dolls.   642  half  dolls.   368  qr.   dolls.   256 
twelve  and  a  half  cent  pieces ;  he  afterwards  gave  a  check 
to  A  for  560  dolls,  and  another  to  B  for  820  dolls. ;  what 
sum  has  he  still  remaining  in  bank?  Ans.  679  dolls. 

17.  A  merchant  bought  a, piece  of  broad  cloth  containing 
«,o  yards,  at  4  dolls.  66  cts.  per  yard ;  of  this  he  found  4 
yards  were  so  damaged  that  he  sold  them  at  half  price  ;  8 
yards  he  sold  at  5  dolls.  50  cents  per  yard :  on  the  whole 
piece  he  gained  IM)  dolls.  56  cents  ;  at  what  rate  did  he  sell 
the  remainder?  Ans.  (>  dollars  per  yard. 

18.  Five  travellers,  upon  leaving  a  tavern  in  I  ho  morning, 
icund  they  were  charged   12£  cents  eaeh    HM-  their  beds,  4 
limes  flint  sum    for  their  spnpor  rind   break  (hst.  75  eenls  fnr 


RKDIJCTJOJV.  39 

liquor  among  them  all,  25  cents  each  ibr  hay  ;  the  remain- 
der of  their  bill,  which  amounted  to  6  dollars,  was  for  oats 
at  2  J  cents  per  quart;  how  many  gallons  of  oats  had  they, 
and  how  much  had  each  man  to  pay  ? 

Ans.  8|  gallons  ;  each  paid  120  cents. 

19.  A  laborer  engaged   to  work  for  75  cents  per  day, 
working  8  hours  each  day,  or  8  hours  for  a  day's  work ;  but 
being  industrious  he  worked  12  hours  25  minutes  each  day 
for  five  days,  and  then  11  hours  30  minutes  for  9  days  more  ; 
what  sum  is  he  entitled  to  receive  for  his  services  ? 

Ans.  15  dollars  52  cents  3  mills-f 

20.  If  25   hhds.  contain  1534  galls.  1   qt.  and   1   pt.  of 
brandy,  each  an  equal  quantity,  how  much  is  there  in  each 
hogshead?  Ans.  61  galls.  1  qt.  1  pt. 

21.  If  a  man  do  114  hours  45  minutes'  work  in  9  days, 
how  long  did  he  work  each  day?  Ans.  I2h.  45mi. 

22.  Divide  180  dollars  among  3  persons  A,  B,  and  C; 
give  B  twice  as  much  as  A,  and  C  three  times  as  much  as  B. 

C  A     20  dolls. 
Ans.  I  B     40 
f  C  120 


SECTION  6. 

OF  REDUCTION. 

REDUCTION  is  the  changing  of  numbers  from  one  denomi- 
nation to  another,  without  altering  their  real  value.  Thus. 
1 -dollar,  if  reduced  to  cents,  will  be  100  cents,  which  in 
their  real  value  are  equal  to  1  dollar:  or,  3  feet  reduced 
to  yards,  is  one  yard,  which  is  still  the  same  length  as  the 
3  feet. 

RULE. 

If  the  reduction  is  from  a  higher  to  a  lower  denomina- 
tion, multiply ;  but  if  from  a  lower  to  a  higher  denomina- 
tion, divide  by  as  many  of  the  next  less  as  make  one  of  the 
greater ;  adding  the  parts  of  the  same  denomination  to  the 
product  as  it  descends ;  and  setting  down  the  remainders  as 
il  ascends. 

Reduction  ascending,  and  descending,  mutually  prove  each 
t.lher. 


40 


REDUCTION. 


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.REDUCTION.  41 

A  TABLE  of  other   Foreign  Coins,  &c.  with  their  value  in  Federal 
Money ,  as  established  by  a  late  act  of  Congress. 

E.  d.  d.  c.  m.  \  E.  d.  d.  c.  m. 

Pound  Sterling'        .     0  4,  4  4  4   t    The  Guilder  of  the 


Pound  of  Ireland    .  0  4,  1   0  0 

Pagoda  of  India      .  0  1,  9  4  0 

Tale  of  China          .  0  1,  2  4  0 

Millree  of  Portugal  0  1,  2  4  0 

Ruble  of  Russia      .  0  0,  6  6  0 

Rupee  of  Bengal     .  0  0,  5  5  5 


U.  Netherlands         0  0,  3  9  0 

Mark  Banco  of  Ham- 
burg 0  0,  3  3  5 

Livre  Turnois  of 

France  0  0,  1  3  5 

Real  Plate  of  Spain     0  0,  1  0  0 


MONEY. 

Cents  are  reduced  to  pence  by  subtracting  one-tenth  of 
their  number.  Pence  are  reduced  to  cents  by  adding  one- 
ninth  of  their  number. 

Pence  are  to  cents  as  9  is  to  10,  and  to  mills  as  9  to  100. 
This  only  applies  where  the  dollar  passes  at  7$.  Qd.  or  90 
pence. 

1.  Reduce  100  cents  to  pence. 

10)  100  cents 
10 

90  pence.  Arts. 

2.  Reduce  90  pence  to  cents. 

9 )  90  pence 
10 

100  cents.  Ans. 

3.  Reduce  1251.  10s.  6%d.  to  farthings. 

125Z.  105. 
20 


2510  shillings 
12 


30126  pence 
4 


120506  farthings.  Ans. 


4.  Reduce  120506  farthings  to  pounds. 

Ans.  1251.  105. 

5.  Reduce  260  cents  to  pence.  Ans.  234d. 

D2 


42  REDUCTION. 

6.  Reduce  480Z.  1 9s.  9d.  to  cents. 

Ans.  128263£  cents. 

7.  Reduce  4658  pence  to  pounds.          Ans.  191.  8s.  2d. 
S.  Reduce  648  pence  to  cents.  Ans.  720  cents. 

9.  Reduce  720  cents  to  pence.  Ans.  648  pence. 

10.  Reduce  24235  half- pence  to  pounds. 

Ans.  507.  9s.  9%d. 

11.  How  many  pounds,  Pennsylvania  currency,  in  216 
French  crowns  ?  Ans.  897.  2s. 

12.  In  29Z.  17s.  how  many  cents  and  dollars? 

A?i».  7960  cents — 79  dolls.  60  cts. 

13.  In  375Z.  Pennsylvania  currency,  how  many  dollars  ! 

Ans.  1000  dolls. 

Note.  To  bring  pounds  (Penn.  currency)  to  dollars,  multiply  by  8  and 
divide  by  3  ;  and  dollars  to  pounds,  multiply  by  3  and  divide  by  8. 

TROY  WEIGHT. 

1.  Reduce  115200  grains  to  pounds.  Ans.  201. 

2.  Reduce  30Z&.  to  grains.  Ans.  172800«r. 

3.  Reduce  45648  pennyweights  to  ounces. 

Ans.  22S2oz.  Sdwt. 

4.  Reduce  4.lb.  Soz.  I5dwt.  20g?\  to  grains. 

Ans.  2726Qgr 

5.  Reduce  27260  grains  to  pounds. 

Ans.  4Z6.  Soz.  15dwt.  20gr. 

6.  In  24  spoons,  each  weighing  Sdwt.  6gr.  how  many 
grains  ?  Ans.  4752gr. 

AVOIRDUPOIS  WEIGHT. 

1.  Reduce  3  tons  to  pounds.  Ans.  6720Z6. 

2.  Reduce  2867200  drams  to  tons.  Ans.  5  tons. 

3.  Reduce  5  tons  to  drams.  Ans.  2867200dfr. 

4.  In  6  barrels  of  flour,  each  weighing  Icwt.  3qr.  Low 
many  pounds?  Ans.  1176/6. 

5.  In  \ficwt.  2qr.  14Z&.  how  many  pounds? 

Ans.  18627A. 

6.  In  a   load  of  hay  weighing  2876Z&.  how  many  hun- 
dreds? Ans.  25cwt.  2qr.  207ft. 

APOTHECARIES1  WEIGHT. 
1.  Reduce  15Z&.  to  scruples.  Ans.  4320*?. 


REDUCTION.  43 

2.  In  a   bottle   containing    3lb.   of  calomel,  how  many 
grains?  Ans.  IT2$Qgr. 

3.  In  2%lb.  of  drugs,  how  many  parcels,  each  16  drams  ? 

Ans.  15  parcels. 

4.  In  576000  grains,  how  many  pounds  ? 

Ans.   100Z&. 

CLOTH  MEASURE. 

1.  Reduce  250  yards  to  nails.  Ans.  4000  nails. 

2.  In  8642  nails,  how  many  Ells  English  ? 

Ans.  432  Ells  E.  2  nails. 

3.  In  324  Ells  French,  how  many  yards  ? 

Ans.  486  yards. 

4.  In  16  bales  of  cloth,  each  measuring  36  Ells  Flemish, 
how  many  yards.?  Ans.  432  yards. 

LONG  MEASURE. 

1.  Reduce  260  miles  to  inches. 

Ans.  16473600  inches. 

2.  Reduce  11  miles  7  furlongs  38  perches  2  yards  2  feet, 
to  barley-corns.  Ans.  2280060&C. 

3.  Reduce  1267200  feet  to  geographical  degrees. 

Ans.  4  degrees. 

4.  Reduce  3  leagues  2  furlongs  110  yards  1  foot  5  inches, 
to  inches.  '•  Ans.  590057  inches. 

5.  How  many  inches  will  reach  round  the  world,  at  60 
miles  to  a  degree?  Ans.  1368576000  inches. 

o 

LAND  MEASURE. 

1.  Reduce  25  acres  to  perches.          Ans.  4000  perches. 

2.  Reduce  176000  perches  to  acres. 

Ans.  1100  acres. 

3.  A  tract  of  land  containing  640000  perches  is  to  be 
iivided  into  400  equal  shares  ;  how  many  acres  will  be  in 

rach  share  ?  Ans.  1 0  acres. 

4.  In  10  acres,  how  many  square  inches  ? 

Ans.  62726400  inches. 


CUBIC,  OR  SOLID  MEASURE. 
1.  Reduce  3200  feet  of  wood  to  cords. 


Ans.  25  cords. 


44  REDUCTION. 

2.  In. 20  tons  of  square  timber,  how  many  feet? 

Ans.  1000  feet. 

3.  In  30  tons  of  round  timber,  how  many  inches  ? 

Ans.  2073600  inches. 

Note.  The  cubic  feet  of  any  circular  body,  such  as  grindstones,  &c 
is  found  in  the  following  manner.  Add  halt'  the  diameter  to  the  whole 
diameter  ;  multiply  the  amount  by  the  aforesaid  half^  and  this  product 
by  the  thickness;  this  will  give  the  contents  in  cubic  inches;  divide 
these  by  1728,  and  the  quotient  will  be  the  cubic  feet. 

4.  In  a  grindstone  48  inches  diameter  and  6  inches  thick, 
how  many  feet  ? 

48  diameter 
24  half  do. 

72 
24 

288 
144 

1728 
6 


1728 )  10368  (  6  cubic  feet.  Ans. 
10368 

5.  In  a  millstone  4  feet  6  inches  diameter,  and  averaging 
18  inches  in  thickness,  how  many  cubic  feet  ? 

54 
27 

81 
27 

567 
162 

2187 
18 


17406 
2187 

1 728 )  39366  ( 22  feet  1350  inches.  Ans. 
3456 

4806 
3456 

1350 


DECIMAL  ARITHMETIC.  46 

TIME. 

1.  Reduce  8  weeks  2  days  6  hours  20  minutes,  to  minutes. 

Ans.  83900  minutes. 

2.  Reduce  ten  years  to  seconds. 

Ans.  315576000  sec. 

3.  How  many  days   since   the    commencement    of    the 
Christian  era  to  the  present  time,  1823 '? 

Ans.  665850  days  18  hours. 

4.  How  many  seconds  in  a  week  ?       Ans.  604800  sec. 

LIQUID  MEASURE. 

1.  Reduce  4  tuns  to  pints.  Ans.  8064  pints. 

2.  Reduce  4032  pints  to  hogsheads.  Ans,  8  hhds. 

3.  Reduce  38  hogsheads  to  pints.         Ans.  19152  pints. 

DRY  MEASURE. 

1.  Reduce  78  bushe's  3  pecks  7  quarts  to  pints. 

Ans.  5054  pints. 

2.  Reduce  2196  pints  to  bushels.     Ans.  346w.  Ipc.  2qt. 


PART  III. 
DECIMAL  ARITHMETIC. 

DECIMAL  ARITHMETIC  is  a  plain  and  easy  method  of  dis- 
covering the  value  of  an  unit,  or  one,  divided  into  any  given 
number  of  parts.  Thus,  if  1  dollar  is  divided  into  10  equal 
parts,  any  one  of  these  parts  will  be  one-tenth,  2  will  be 
two-tenths,  3,  three-tenths,  &c.  Again,  if  1  dollar  is  divided 
into  a  hundred  equal  parts,  any  one  of  these  will  be  one- 
hundredth,  2,  two-hundredths,  &c. 

The  number  of  parts  into  which  the  unit  is  divided  is 
called  the  denominator,  and  any  number  of  these  parts  less 
than  the  whole  is  called  the  numerator,  and  which  always 
stands  over  the  denominator  ;  thus, 
2  numerator 
10  denominator 

is  read  two-tenths  ;  and  these  two  so  placed  constitute 
what  is  termed  a  fraction.  In  decimal  fractions,  the  de- 
nominator can  only  be  an  unit,  with  one  or  more  ciphers 


46  ADDITION  OF  DECIMALS. 

added  thereto  ;  as  y50-,  y2^,  T2/FV  The  numerators  of  these 
are  usually  written  without  their  denominators,  and  are  dis- 
tinguished from  whole  numbers,  by  prefixing  a  point  called 
the  separatrix,  as  ,5  ,25  ,'225. 

Ciphers  placed  to  the  right  hand  of  decimals,  make  no 
change  in  their  value,  for  ,5  ,50  ,500,  &c.  are  decimals  of 
the  same  value,  each  being  equal  to  J.  But  when  prefixed 
to  the  decimal,  they  decrease  the  value  in  a  tenfold  propor- 
tion. Thus,  ,5  ,05,  ,005,  have  the  same  proportion  to  each 
other  a,s  5,  50,  500,  have  in  whole  numbers. 

This  is  made  plain  by  the  following 

TABLE. 
Integers.  Decimals. 


5 

5  0  ,5 

500  ,05 

5000  ,005 

50000  ,0005 

500000  ,00005 

5000000  ,000005 


all's  11 

.2  13   w  3  "2  <»  ,£2 

=3    5    G    Q    5    C  -^ 


SECTION  I. 

ADDITION  OF  DECIMALS. 

RULE. 

SET  down  the  numbers  according  to  their  value,  viz. — 
units  under  units,  tenths  under  tenths,  &c.  Then  add  as 
in  addition  of  whole  numbers,  and  place  the  point  in  the 
amount  exactly  und^i  those  in  the  given  sum. 


SUBTRACTION  OF  DECIMALS  47 

EXAMPLES. 

2468,5036  3460000,0000643 

521,0428  460000,000643 

32,0004  3400,3680005 


3021,5468 


3.  Add     283,604  +  490006,003275  -f  21,05  -f  1,2  + 
6200,3476.  Ans.  496512,204875. 

4.  Add  ,246 +  ,012 +  ,02 +  ,6 +  ,41 3 +  ,5.      Ans.  1,791. 

5.  Add     25,52  +  225,005 +  ,0035 +  844 -f2,2 +  300,825 
+  ,00005.  Ans.  1397,55355. 

6.  Add  one  hundred  and  twenty-five,  and  five-tenths,  + 
ten  thousand,  and  five  millionths,  + fifteen,  and  seventy-two 
thousandths,  +  two,  and  one  hundredth. 

A.ns.  10142,582005. 

7.  Add  five,  and  four-tenths,  +  fifteen  and  four  hundredths, 
+  one  hundred,  and  four  thousandths,  +  six  thousand  and 
four    hundred   thousandths,  +  ninety-three    thousand    eight 
hundred  and  eighty,  and  four  ten  thousandths. 

Ans.  100000,44444. 


SECTION  2. 
SUBTRACTION  OF  DECIMALS. 

RULE. 

SET  the  less  under  the  greater,  with  the  points  as  in  addi- 
tion, and  place  the  point  in  the  remainder,  in  the  same 
manner. 

EXAMPLES. 

Prom  6432,50437  From  848,045  From  15,6547 

Take     369,95429          Take  162,54936^        Take     7,35 


Kern.  6062,55008 


4.  From  45,005  take  23,65^82.  Ans.  21,35018. 

5.  From  six  hundred  and  twenty,  and  two-tenths,  take 
wo  hundred  and  two  thousandths.  Ans.  420,li/krt 


48  MULTlPLICATIGiN  OF  DECIMALS. 

6.  From  5  take  ,10438.  Ans.  4,89562. 

7.  From  2  take  ,00002.  Ans.  1,99998. 

8.  From  sixteen  take  sixteen  thousandths  parts. 

Ans.  15,984. 


SECTION  3. 

MULTIPLICATION  OF  DECIMALS. 

RULE. 

MULTIPLY  as  in  whole  numbers,  and  from  the  product 
point  ofF  as  many  on  the  right-hand  for  decimals,  as  there 
are  in  both  the  factors.  If  the  whole  product  should  be  too 
few,  then  must  ciphers  be  added  on  the  left  of  the  product 
till  an  equal  number  is  had. 


EXAMPLES. 


Multiply  29,831 
by       ,952 


59662 
149155 
268479 

Product       28,399112 


24,021 
4,23 

22,2043 
,12345 

72063 
48042 
96084 

1110215 

888172 
666129 
444086 
222043 

101,60883 

4.  Multiply  ,385746  by  ,00463 

5.  158,694  23,15 

6.  ,024653         ,00022 


2,741120835 

Ans.  ,00178600398 

3673,7661 

,00000542360. 


7.  Multiply  twenty-five  and  four  hundredths,  by  two  thou- 
sandths. Ans.  ,05008. 

8.  Multiply  six  hundred  and   forty-five,  and  three  thou- 
sandths, by  five  millionths.  Ans.  ,003225015. 

Note.  The  product  of  any  number  when  multiplied  by  a  decimal 
only,  will  be  less  than  the  multiplicand,  in  llrt  PUIIW  proportion  as  the 
multiplier  is  les«  than  one. 


MULTIPLICATION  OF  DECIMALS.  49 

CONTRACTION  IN  MULTIPLICATION  OF  DECIMALS. 

If  only  a  limited  number  of  decimals  is  sought  for,  instead 
of  retaining  the  whole  product,  obtained  by  the  foregoing 
method,  work  by  the  following 

RULE. 

1.  Set  the  multiplier,  in  an  inverted  order,  under  thn 
multiplicand,  placing  the  units  figure  of  the  multip'ier  under 
the  lowest  decimal  place  in  the  multiplicand,  that  is  wished 
to  be  retained. 

2.  In  multiplying,  omit  those  figures  in  the  multiplicand 
which  are  on  the  right  of  the  multiplying  figure,  but  to  the 
first  figure  in  each  line  of  the  product,  add  the  carriage 
which  would  arise  from  the  multiplication  of  the  omitted 
figures,  carrying  one  from  5  to  15,  2  from  15  to  25,  3  fron 
25  to  35,  &c.     Place  the  first  figures  in  each  product  di- 
rectly under  each  other,  and  add  as  in  addition. 

Note.  If  you  would  be  absolutely  certain  that  the  last  figure  retained 
is  the  nearest  to  the  truth,  work  for  one  place  more  than  you  wish  to 
retain* 

EXAMPLES. 

1.  Multiply  34,6733  by  3,1416,  retaining  four  decimal 
places  in  the  product. 

34,6733 
61413  inverted 


1040199 
34673 
13869 

347 

209 

108,9296  Arts. 

2.  Multiply   ,78543  by  ,346787,  retaining  five  decimal 
places  in  the  product. 

34G787  Or  thus,  ,78543 

34587      inverted  787643,0 

24275  235G3 

2774  3142 

173  471 

14  55 

1  6 


.27237    4ns  427237 


50  DIVIS[OIN  OF  DECIMALS. 

3.  Multiply  23,463  by  2,34,  retaining  three  decimals. 

Ans.  54,903. 

4.  Multiply  234,216  by  2,345,  retaining  two  decimals. 

Ans.  549,23. 

5.  Multiply   3,141592  by  52,7438,  retaining  four  deci 
mals.  Ans.  165,6995, 


SECTION  4 
DIVISION  OF  DECIMALS. 

RULE. 

DIVIDE  in  the  same  manner  as  in  whole  numbers,  and 
point  off  on  the  right  of  the  quotient  as  many  figures  for  de- 
cimals, as  the  decimals  in  the  dividend  exceed  those  in  the 
divisor.  When  the  decimals  in  the  divisor  exceed  those  in 
the  dividend,  let  ciphers  be  added  to  the  dividend,  till  they 
equal  those  in  the  divisor.  And  if  there  be  a  remainder,  let 
ciphers  be  annexed  thereto,  and  the  quotient  carried  on  to 
any  degree  of  exactness. 

EXAMPLES. 

29,831  )  2'  ,d991 12  (  ,952       24,021  )  101,60883  (  4,28 
'  268479  96084 


155121  55248 

149155  48042 


59662  72063 

59662  72063 


Nate.  When  the  divisor  is  10,  100, 1000,  &c.  the  division  is  perform- 
ed by  pointing  off  as  many  figures  in  the  dividend  for  decimals,  aa 
there  are  ciphers  in  the  divisor. 

10  )  I  685,6 

Thus,  6856  divided  by          100  >      is  <  68,56 

1000  >  (  6,856 

3.  Divide  65321  by  23,7  Ans  2756,16  + 

4.  234,70525  64,25  3,653 

5.  10  3  3,3333  + 

6.  9  ,9  10 

7.  ,00178600398  ,00463  ,385746 


DIViSIOiN  OF  DECIMALS.  fj  1 

8.  Divide  ,2327898         by         2,46  Ans.  ,09463 

9.  '  ,2327898  ,09463  2,46 
10.         *      ,000162                         ,018  ,009 

CONTRACTION  IN  DIVISION  OF  DECIMALS. 

When  only  a  limited  number  of  decimals  in  the  quotient 
is  sought  for,  work  by  the  following 
RULE. 

1.  Take  as  many  figures  only  on  the  left  hand  side  of 
the  divisor,  as  the  whole  number  of  figures  sought  for  in  the 
quotient,  and  cut  off  the  rest. 

2.  Make  each  remainder  a  new  dividend,  and  for  a  new 
divisor,  point  off  one  figure  continually  from  the  right  hand 
of  the  former  divisor,  taking  care  to  bring  in  the  increase, 
or  carriage  of  the  figures  so  cut  off,  as  in  multiplication. 

Note.  When  the  whole  divisor  does  not  contain  as  many  figures  as 
are  sought  for  in  the  quotient,  proceed  as  in  common  division,  without 
cutting  off  a  figure,  till  the  figures  in  the  divisor  shall  equal  the  re- 
maining figures  required  in  the  quotient,  and  then  begin  to  cut  off  as 
above  directed. 

EXAMPLES. 

1.  Divide  14169,206623851  by  384,672258,  retaining 
four  decimal  places  in  the  quotient,  or  in  all  six  quotient 
figures. 

3.8.4,6.7.2|258  )  14169,206623851  (  36,8345.  Ans. 
1154017 


262903 
230803 

32100 
30774 


1326 
1154 

172 
153 

19 
19 


52  REDUCTION  OF  DECIMALS. 

2.  Divide  ,07567  by  2,32467,  true  to  four  decimal  places, 
or  three  significant  figures,  the  first  being  a  cipher. 

2,3.2|467  )  ,07567  (  ,0326.  Ans. 
697 

59 
46 

13 
14 

3.  Divide  5,37341  by  3,74,  true  to  four  decimal  places. 

3,74  )  5,37341  ( 1 ,4367.  Ans. 
374 

1633 
1496 

1374 
1122 


252 
224 

28 
26 

4.  Divide  74,33373  by  1,346787,  true  to  three  decimal 
places.  Ans.  55,193. 

5.  Divide  87,076326  by  9,365407,  true  to  three  decimal 
places.  Ans.  9,297. 

6.  Divide  32,68744231   by  2,45,   true   to   two   decimal 
olaces.  Ans.  13,34. 

7.  Divide  ,0046872345   by  6,24,  true   to   five   decima 
places.  Ans.  ,00075 


SECTION  5. 
REDUCTION  OF  DECIMALS. 

Case  1. 
To  reduce  a  vulgar  fraction  to  a  decimal. 


REDUCTION  OF  DECIMALS.  53 

RULE. 

Annex  one  or  more  ciphers  to  the  numerator,  and  divide 
by  the  denominator;  the  quotient  will  be  the  answer  in 
decimals. 

EXAMPLE. 

1.  Reduce  £  to  a  decimal. 

4)1,00 


,25     Ans. 

2.  Reduce  £  to  a  decimal.  Ans.  ,5 

3.  -  J  to  a  decimal.  ,75 

4.  -  -J  to  a  decimal.  ,875 

5.  -  -Jj  to  a  decimal.  ,04 

6.  -  |j  to  a  decimal.  ,95 

7.  -  T63  of  a  dollar  to  cents.  ,40  cts. 

Case  2. 

To  reduce  numbers  of  different  denominations  to  a  deci- 
mal of  equal  value. 

RULE. 

Set  down  the  given  numbers  in  a  perpendicular  column, 
having  the  least  denomination  first,  and  divide  each  of  them 
by  such  a  number  as  will  reduce  it  to  the  next  name,  annex- 
ing  the  quotient  to  the  succeeding  number  ;  the  last  quotient 
will  be  the  required  decimal. 

EXAMPLE. 

1.  Reduce  17s.  8%d.  to  the  decimal  of  a  pound. 


4 
12 
20 


3 

8,75 
17,720166 

,8864583  +   Ans. 


2.  Reduce  195.  to  the  decimal  of  a  pound.       Ans.  ,95 

3.  -        3d.  to  the  decimal  of  a  shilling.  ,25 

4.  -        3d.  to  the  decimal  of  a  pound.  ,0125 

5.  -        kcwt.  2qr.  to  the  decimal  of  a  ton.  ,225 

6.  -         2qr.  14/6.  to  the  decimal  of  a  cwt.  ,625 

7.  -         3qr.  3na.  to  the  decimal  of  a  yard.  ,9375 

E2    • 


54  REDUCTION  OF  DECIMALS. 

Case  3. 
To  reduce  a  decimal  to  its  equal  value  in  integers. 

RULE. 
Multiply  the  decimal  by  the  known  parts  of  the  integer 

EXAMPLE. 

1.  Reduce  ,8864583  of  a  pound  to  its  equivalent  value 
in  integers. 

,8864583 
20 


s.  17,7291660 
12 

d.  8,7499920 
4 

qr.  2,9999680 

It  is  usual  when  the  left  hand  figure  in  the  remaining 
decimal  exceeds  five,  to  expunge  the  remainder,  and  add 
one  to  the  lowest  integer.  Thus,  instead  of  17s.  8d.  2,999, 
die.  we  may  say  17s.  8%d.  Ans. 

2.  What  is  the  value  of  ,75  of  a  pound  ?          Ans.  15s. 

3.  What  is  the  value  of  ,7  of  a  pound  troy? 

Ans.  Soz.  Sdwt. 

4.  What  is.  the  value  of  ,617  of  a  cwt.  ? 

Ans.  2qr.  13Z&.  1  oz.  W  +  dr. 

5.  What  is  the  value  of  ,3375  of  an  acre? 

Ans.  I  rood,  14per. 

6.  What  is  the  value  of  ,258  of  a  tun  of  wine  ? 

Ans*  Ihhd.  2 -{-gals. 

7.  What  is  the  proper  quantity  of  ,761  of  a  day? 

Ans.  ISh.  15?ni.  50,4src. 

i.  What  is  the  proper  quantity  of  ,7  of  a  Ib.  of  silver? 

Ans.  Soz.  Sdwt. 

:•    What  is  the  proper  quantity  of  ,3  of  a  year? 

Ans.  lQ9d.  13ft 

10.  What  is  the  difference  between  ,41  of  a  day  and  ,16 
of  an  hour  ?  Ans.  97*.  40wi.  48sec. 

11.  What  is  the  sum  of  ,17T.  19™^.  ,l7qr.  and  lib.  1 

Ans.  Zcwt.  2qr. 


DECIMAL  KHACT1O.NS.  55 

Promiscuous  Questions  in  Decimal  Fractions. 

1.  Multiply  ,09  by  ,000.  Ans.  ,00081. 

2.  In  ,36  of  a  ton  (avoirdupois)  how  many  ounces  ? 

Ans.  12902,402. 

3.  What  is  the  value  of  ,9125  of  an  ounce  troy? 

Ans.  18dwt.  6gr. 

4.  Reduce  ^j  to  a  decimal.  Ans.  ,0127  nearly.  . 

5.  Reduce  2oz.  I6dwt.  20gr.  to  the  decimal  of  a  pound 
troy.  Ans.  ,2368 -f  nearly. 

6.  What  is  the  length  of  ,1392  of  a  mile? 

Ans.  1  fur.  4  per.  3  yds  nearly. 

7.  What  multiplier  will  produce  the  same  result,  as  mul- 
tiplying by  3,  and  dividing  the  product  by  4  ? 

Ans.  ,75. 

8.  What  decimal  of  Icwt.  is  6lb.  Ans.  ,0535714. 

9.  What  part  of  a  year  is  109  days  12  hours? 

Ans.  ,3. 

10.  In  ,04  of  a  ton  of  hewn  timber,  how  many  cubic 
inches?  Ans.  3456. 

11.  What  is  the  value  of  T3T  of  a  dollar  divided  by  3? 

Ans.  6f  cents. 

12.  What  is  the  value  of  ,875  of  a  hhd.  of  wine? 

Ans.  55  gal.  0  qt.  1  pt. 

13.  What  divisor,  true  to  six  decimal  places,  will  produce 
the  same  result  as  multiplying  by  222  ? 

Ans.  ,004504. 

14.  In  ,05  of  a  year,  how  many  seconds,  at  365  days  6 
hours  to  the  year?  Ans.  1577880. 

15.  What  number  as  a  multiplier  will  produce  the  same 
result  as  multiplying  by  ,73  and  dividing  first  by  3,  and  the 
quotient  by  ,25  ?"  Ans.  ,973^. 

16.  What  is  the  difference  between  ,05  of  a  year,  and  ,5 
of  an  hour  ?  Ans.  2i».  2d~  ISh.  42m. 

17.  In  ,4  of  a  ton,  ,3  of  a  hhd.  and  ,8  of  a  gallon,  how 
many  pints  ?  Ans.  964. 

18.  How  many  perches  in  ,6  of  an  acre;  multiplied  by 
X)2?  Ans.  1,92.* 

19.  What  part  of  a  cord  of  timber  is  1  cubic  inch? 

Ans.  ,000004  -f 

20.  What  part  of  a  circle  is  28  deg.  48  minutes  ? 

ATU.  ,08. 


66  SINGLE  RULE  OF  THREE  DIRECT 

PART   IV. 

PROPORTIONS. 

Tins  part  of  arithmetic  which  treats  of  proportions  is 
very  extensive  and  important.  By  it  an  almost  innumerable 
variety  of  questions  are  solved.  It  is  usually  divided  into 
three  parts,  viz.  Direct,  Inverse,  and  Compound. — The  first 
of  these  is  called  the  Single  Rule  of  Three  Direct,  and 
sometimes  by  way  of  eminence  the  Golden  Ride.  The  sec- 
ond is  called  the  Single  Rule  of  Three  Inverse:  and  the 
last  is  called  the  Double  Rule  of  Three.  In  all  these,  cer- 
tain numbers  are  always  given,  called  data*  by  the  multipli- 
cation and  division  of  which,  the  answer  in  an  exact  ratio 
of  proportion  to  the  other  terms  is  discovered. 


SECTION  1. 

SINGLE  RULE  OF  THREE  DIRECT 

IN  this  rule  three  numbers  are  given  to  find  a  fourth,  that 
shall  ha\e  the  same  proportion  to  the  third,  as  the  second 
has  to  the  first. 

If  by  the  terms  of  the  question,  more  requires  more,  or 
less  requires  lessy  it  is  then  said  to  be  direct,  and  belongs  to 
this  rule. 

In  stating  questions  in  this  rule,  the  middle  term  must 
always  DC  of  the  same  name  with  the  answer  required;  the 
last  term  is  that  which  asks  the  question,  and  that  which  is 
of  the  same  name  as  the  demand,  the  first.  When  the  ques- 
tion is  thus  stated,  reduce  the  first  and  third  terms  to  the 
lowest  denomination  in  either ;  and  the  middle  term  (if  com- 
pound) to  its  lowest,  and  proceed  according  to  the  following 

RULE. 

Multiply  the  second  and  third  terms  together,  and  divide 
the  product  by  the  first ;  the  quotient  will  be  the  fourth  term, 
or  answer,  in  the  same  name  with  the  second. 


SINGLE  RULE  OF  THREE  DIRECT.  5? 

PROOF. 

Invert  the  question,  making  the  answer  the  first  term ;  the 
result  will  be,  the  first  term  in  the  original  question. 

Note.  1.  After  division  if  there  be  any  remainder,  and  the  quotient 
be  not  in  the  lowest  denomination,  it  must  be  reduced  to  the  next  less 
denomination,  dividing  as  before,  till  it  is  brought,  to  the  lowest  denomi- 
nation, or  till  nothing"  remains. 

2.  When  any  of  the  terms  are  in  federal  money,  the  operation  is  con- 
ducted  in  all  respects  as  in  simple  numbers,  taking  care  to  place  the 
separatrix  bet'ween  dollars  and  cents,  according  to  what  has  already 
been  laid  down  in  federal  money  and  decimal  fractions. 

EXAMPLE. 

1.  If  8  yarcls  of  cloth  cost  32  dollars,  what  will  24  yards 
cost  ? 

Yds.       D.        Yds.  D.        Yds.        D. 

As    8     :     32     ::     24  Proof.     As  96     :     24     ::     32 

24  32 

128  48 

64  72 

8)768  96)768(8 

768 
96  Ars.  

2.  When  sugar  is  sold  at  12  dollars  32  cts.  per  cwt.  whni 
will  Wlb.  cost?  Ans.  I  doll.  76  cts. 

3.  What  is  the  amount  of  3  cwt.  of  coffee  at  36  cents  per 
pound?  Ans.  120  dolls.  96  cts. 

4.  What  will  4  pieces  of  linen  come  to,  containing  23, 
24,  25,  and  27  yards,  at  72  cents  per  yard  ? 

Ans.  71  dolls.  28  cts. 

5.  What  will  bcwt.  2qr.  Sib.  of  iron  come  to  at  48  cents 
for  4/6.  ?  Ans.  61  dolls.  44  cts. 

6.  What  will  V2Slb.  of  pork  come  to  at  8  cts.  per  pound? 

Ans.  10  dolls.  24  cts. 

7.  If  9£  dozen  pair  of  stockings,  cost  68  dollars  40  cent? 
what  will  3  pair  cost  ?  Ans.  1  doll.  80  cts. 

8.  If  20  bushels  of  oats  cost  9  dollars  60  cents,  what  will 
three  bushels  come  to?  Ans.  1  doll.  44  cts. 

9.  A  merchant  bought  a  piece  of  cloth  for  16  dollars  50 
cents,  at  75  cents  per  yard  ;  how  many  yards  were  there  iii 
the  piece  ?  Ans.  22  yds. 

10.  If  17 cwt.  3qr.  17  Ib.  of  sugar  cost  320  dollars  80  cts. 
what  must  be  paid  for  6oz.  ?  An*.  6  o*nts. 


58  SINGLE  RULE  OF  THREE  DIRECT 

11.  If  9,7 Ib.  of  silver  is   worth   97   dollars,  what  is  the 
value  of  l,5oz. 1  Ans.  1  doll.  25  cts. 

12.  If  125,5  acres  are  sold  for  627,5  dollars,  what  will 
4,75  acres  cost  ?  Ans.  23  dolls.  75  cts. 

13.  If  1,5  gallons  of  wine  cost  4  dollars  50  cents,  what 
will  1,5  tuns  cost?  Ans.   1134  dolls. 

14.  How  many  reams  of  paper  at  1  dollar  66  cents,  1 
dollar  97  cents,  and  2  dollars  31  cents  per  ream  may  be 
purchased  for  528  dollars  66  cents,  of  each  an  equal  num- 
ber ?  Ans.  89  reams  of  each  sort. 

15.  When  iron  is  sold  for  224  dollars  per  ton,  what  will 
Iqr.  1Mb.  cost?  Ans.  4* dolls.  20  cts._ 

16.  A  merchant  paid  1402  dollars  50  cents  for  flour,-  at  5 
dollars  50  cents  per  barrel ;  how  many  barrels  must  he  re- 
ceive? Ans.  255  barrels. 

17.  A  man  has  a  yearly  salary  of  1186  dollars  25  cents, 
how  much  is  it  per  day  ?  Ans.  3  dolls.  25  cts. 

18.  A  man  spends  2  dollars  25  cents  per  day,  and  saves 
378  dollars  75  cents  at  the  end  of  the  year,  what  is  his 
yearly  salary?  Ans.  1200  dolls. 

19.  What  will  4T.  Wcwt.  Iqr.  I2lb.  of  hay  come  to  at 
1  dollar  12  cents  per  cwt.?  Ans.  101  dolls.  20  cts. 

20.  How  much  will  a  grindstone  4  feet  6  inches  diameter, 
stfid  9  inches  thick,  come  to  at  1  dollar  10  cents  per  cubic 
foot?  Ans.  12  dolls.  53  cts. 

21.  \\hat  will  a  grindstone  28  inches  diameter,  and  3,5 
inches  thick,  come  to  at  1  dollar  90  cents  per  cubic  foot  ? 

Ans.  2  dolls.  26  cts. 

22.  At  221.  Ss.  per  ton,  what  will  203  T.  Scwt.  3qr.  Mb. 
of  tobacco  come  to  ?  Ans.  4558Z.  3s. 

23.  If  850  dolls.  50  cents  is  paid  for  18  pieces  of  cloth 
at  the  rate  of  11  dollars  25  cents  for  5  yards,  how  many 
yards  were  in  each  piece,  allowing  an  equal  number  to  each 
piece?  Ans.  21  yds. 

24.  If  124  yards  of  muslin  cost  1Z.  17$.  6rf.  what  is  it 
pe>  yard?  Ans.  3s. 

25.  If  a  staff  4  feet  long  cast  a  shadow  (on  level  ground) 
7  feet  long,  what  is  the  height  of  a  steeple  whose  shade  at 
the  same  time,  is  218  feet  9  inches?  A?is.  125  feet. 

26.  If  4292  dollars  32£  cents  are  paid  for  476  acres  3 
roods  28  perches  of  land,  how  much  is  it  per  acre  ? 

Ans.  9  dollars. 

27.  If  a  man's  annual  income  be  1333  dollars,  and  he 


SLN(iLK  lUiJJ-:  OK  TUkiaO  INVERSK.  59 

expend  daily  2  dollars  1-4  cents,  how  much  will  lie  save  at 
'the  end  of  the  year  1  Ans.  551  dolls.  90  cts. 

28.  If  321    bushels  of  wheat  cost  240  dollars  75  cents, 
what  is  it  per  bushel  ?  Ans.  75  cts. 

29.  If  l£  yard  of  cloth  cost  2  dollars  50  cents,  what  will 
1  quarter  2  nails  come  to  ?  Ans.  62^  cts. 

30.  Bought  3  pipes  of  wine,  containing  120-|,  124,  and 
126|  gallons,  at  5s.  6d.  per  gallon ;  what  do  they  cost  ? 

Ans.  102Z.  Is.  lOicZ. 

31.  A  sets  out  from  a  certain  place  and  goes  12  miles  a 
day ;  5  days  after,  B  sets  out  from  the  same  place,  the  same 
way,  and  goes  16  miles  a  day;  in  how  many  days  will  he 
overtake  A?  Ans.  15  days. 

32.  If  I  have  owing  to  me  1000Z.  and  compound  with  my 
debtor,  at,  12s.  Qd.  per  pound,  how  much  must  I  receive? 

Ans.  625Z. 

33.  If  365  men  consume  75  barrels  of  pork  in  9  months, 
how  many  will  500  men  consume  in  the  same  time  ? 

Ans.   10241  barrels. 

34.  How  much  land  at  2  dollars  50  cents  per  acre,  musl 
f>e  given  in  exchange  for  360  acres  at  3  dollars  75  cents  1 

Ans.  540  acres. 

35.  If  the  earth,  which  is  360  degrees  in  circumference, 
turns  round  on  its  axis  in  24  hours,  how  far  are  the  inhabit- 
ants at  the  equator  carried  in  1  minute,  a  degree  there  being 
69!  miles  ?  Ans.  17  miles  3  fur. 


SECTION    2. 

SINGLE  RULE  OF  THREE  INVERSE. 

IF  in  any  given  question,  more  requires  less,  or  less  re- 
quires more,  the  proportion  is  inverse,  and  belongs  to  this 
rule. 

Having  stated  the  question,  as  in  the  rule  of  three  direct, 
proceed  according  to  the  following 

RULE. 

Multiply  the  first  and  second  terms  together,  and  divide 
the  product  by  the  third ;  the  quotient  will  be  the  answer,  in 
the  same  name  as  the  second. 


60  SINGLE  RULE  OF  THREE  INVERSE 

EXAMPLE. 

1.  If  20  men  can  build  a  wall  in  12  days,  how  long  will 
it  require  40  men  to  build  the  ^ame  ? 

M.          d.  M.  d.         M.  d. 

As  20  :  12  ::  40     Proo/  As  6  :  40  ::  12 
12  40 


40  )  240  (  6  days.  Ans.          12  )  240  ( '20  men. 
240  24 


0 

2.  If  60  men  can  build  a  bridge  in  100  days,  how  long 
will  it  require  20  men  to  build  it  ?  Ans.  300  days. 

3.  If  a  wall  100  yards  long  requires  65  men  4  days,  in 
what  time  would  5  men  complete  it  ?  Ans.  52  days. 

4.  If  a  barrel  of  flour  will  last  a  family  of  six  persons  24 
days,  how  long  would  it  last  if  3  more  were  added  to  the 
family?  Ans.   16  days. 

5.  If  5  dollars  is  paid  for  the  carriage  of  \cwt.  weight, 
150  miles,  how  far  may  Sent*  weight  be  carried   for  th^ 
same  money  ?  Ans.  25  miles. 

6.  If  a  street  80  feet  wide  and  300  yards  long,  can  be 
paved  by  40  men  in  20  days,  what  length  will  one  of  60 
feet  wide  be  paved  by  the  same  men  in  the  same  time  ? 

Ans.  400  yards. 

7.  If  a  field  that  is  30  rods  wide  and  80  in  length,  con- 
tain  15  acres,  how  wide  must  one  be  to  contain  the  same 
quantity,  that  is  but  70  rods  long  ?     Ans.  3412.  4/7.  8&iri. 

S.  If  a  board  be  ,75  of  a  foot  wide,  what  length  must  it 
be  to  measure  12  square  feet?  Ans.  16  feet. 

9.  How  much  cloth   1,25   yards  wide,  can  be  lined  by 
42,5  yards  of  silk  that  is  ,75  of  a  yard  wide  ? 

Ans.  25,5  yards. 

10.  If  10  men  could  complete  a  building  in  4,5  months, 
what  time  would  it  require  if  5  more  were  employed  ? 

Ans.  3  months. 

11.  In  what  time  will  600  dollars  gain  50  dollars,  when 
80  dollars  would  gain  it  in  15  years?  fins.  2  years. 

12.  If  a  traveller  can  perform  a  journey  in  4  days,  when 
the  days  are  12  hours  long,  what  time  will  lw  require  when 
thf!  days  arn  1  (>  hours  long  ?  _  Ans.   3  days. 

I  ;*.   Suppose   100   nvn   in   a   garrison    are   supplied 


SINGLE  RULE  Oi    TliUKE  INVERSE.  61 

provisions  for  30  days,  how  many  men  must  be  sent  out  if 
they  would  have  the  provisions  last  50  days  ? 

Ans.  160  men. 

14.  Lent  a  friend  292  dollars  for  six  months  ;  afterwards 
J  borrow  from  him  806  dollars  ;  how  long  may  I  keep  it  to 
balance  the  favor?  Ans.  2  months  5  days. 

15.  1200  men  stationed  in  a  garrison,  have  provisions  for 
9  months,  at  the  rate  of  14  ounces  per  day;  how  long  at 
the  same  allowance  will  the  same  provisions  last  if  they  are 
reinforced  by  400  men  ?    And  also  what  diminution  must  be 
made  on  each  ration,  that  the  provisions  may  last  for  the 
same  time  ?  Ans.  6|  mo.  at  the  same  allowance — 

3J  oz.  deduction  to  last  for  the  same  time. 

16.  If  a  piece  of  land  40  rods  in  length  and  4  in  breadth, 
make  an  acre,  how  wide  must  it  be  if  it  is  but  25  rods  long? 

•     Ans.  6|  rods. 

17.  How  much  in  length  that  is  3  inches  broad,  will  make 
a  square  foot  ?  Ans.  48  inches. 

18.  If  a  pasture  field  will  feed  6  cows  91  days,  how  long 
will  it  feed  21  cows  ?  Ans.  26  days. 

19.  There  is  a  cistern  having  1  pipe,  which  will  empty  it 
in  10  hours ;  how  many  pipes  of  the  same  capacity  will 
empty  it  in  24  minutes  ?  Ans.  25  pipes. 

20.  How  many  yards  cf  carpeting  that  is  half  a  yard 
wide,  will  cover  a  floor  that  is  30  feet  long  and  18  feet  wide? 

Ans.  120  yards. 

21.  What  is  the  weight  of  a  pea  to  a  steelyard,  which 
being  suspended  39  inches  from  the  centre  of  motion,  will 

l 

Ans.  4Z6. 

22.  A  and  B  depart  from  the  same  place,  and  travel  the 
same  road  ;  but  A  goes  5  days  before  B  at  the  rate  of  20 
miles  a  day,  B  follows  at  the  rate  of  25  miles  a  day ;  in 
what  time,  and  at  what  distance,  will  he  overtake  A? 

Ans.  20  days,  and  500  miles. 

The  following-  rule,  if  adopted,  will  suit  for  the  stating  of  all  questions 
in  single  proportion,  whether  direct  or  inverse. 

GENERAL  RULE, 

Place  that  number  for  the  third  term,  which  signifies  the 
same  kind,  or  thing,  as  that  which  is  sought ;  arid  consider 
whether  the  number  sought  will  be  greater  or  less  ;  if  great- 

F 


equipoise  2QSlb.  suspended  at  the  draught  end  f  of  an  inch  ? 


f>2  SLXGLK  RULF.  OF  THRRK. 

er,  place  the  least  of  the  other  terms  for  the  first,  but  if  less 
place  the  greater  for  the  first  term,  and  the  remaining  one 
for  the  second. 

Multiply  the  second  and  third  terms  together,  and  divide 
the  product  by  the  first ;  the  quotient  will  be  the  answer 
required. 

EXAMPLES. 

1.  If  30  horses  plow  12  acres,  how  many  will  40  horses 
plow  in  the  same  time  ? 

k.  h.  acr. 

Direct  Proportion.      30      :      40      :  :      12 

12 


30)480(16  acres.      A  us. 

2.  If  30  horses  plow  12  acres  in  10  days,  in  how  many 
days  will  40  horses  plow  the  same  quantity  ? 
h.  h.  D. 

Inverse  Proportion.     40     :     30     : :     10 

10 


40)300(7,5  days.     Ans. 

3.  If  800  soldiers  in  a  garrison  have  provisions  sufficient 
for  2  months  ;  how  many  must  depart  that  the  provisions 
may  last  them  for  5  months  ?  Ans.  480. 

1.  Bought  a  hogshead  of  Madeira  wine  for  119  dollars, 
nine  gallons  of  which  leaked  out ;  what  was  the  remainder 
sold  at  per  gallon,  to  gain  12  dollars  on  the  whole  1 

-Ans.  2  dolls.  42-J-fcts. 

5.  If  225  pounds   be  carried  51*3  miles  for  20  dollars, 
how  many  pounds  may  be  carried  64  miles   for   the  same 
money?  Ans.  1800Z6. 

6.  If  87  dolls.  50  cents  be  assessed  on  1750  dolls,  what 
is  the  tax  of  10  dolls,  at  the  same  rate?  Ans.  50  cts. 

Promiscuous  Questions  in  Direct  and  Inverse  Proportion. 

1.  Suppose  a  man  tnivels  to  market  with  his  wagon 
loaded,  at  the  rate  of  2-£  miles  «.n  hour,  and  returns  with  \\ 
empty  at  the  rate  of  3£  miles  an  hour  ;  how  long  will  he  be 
in  performing  a  journey,  going  and  returning,  to  a  place 
123  miles  distant?  Ana.  84J-J  hoars. 


SINGLE  HULK  OF  TlfREK.  (>3 

•2.   A  lent  B  1000  dollars   fur  IS!)  days,  without   interest 
how  long  should  B  'end  A  iif>0  dollars  to  requite  the  favor? 

Ans.  290fg  days. 

3.  Bought   14  casks  of  butter,  each  weighing  Icwt.  Iqr. 
4/6.  at  12  dollars  60  cents  per  cwt. ;  what  did  they  come  to, 
and  how  much  per  Ib.  ? 

Ans.  226  dolls.  80  cts.  whole  cost ;  1 1  cts  2 \  m.  per  Ib. 

4.  Sold  4  chests  of  tea,  each  weighing  Icwt.  Qqr.  14//>. 
the  first  for  80  cents  per  Ib.  the  second  for  90  cents,  the 
third  for  1  doll.  5  cents,  and  the  fourth  for  1  doll.  25  cents  ; 
how  many  pounds  of  tea  were  there,  what  was  the  average 
price,  and  what  did  the  whole  come  to  ? 

Ans.  504//?. — average  1  doll. — come  to  504  dolls. 

5.  When  flour  is  sold  at  2  dolls.  24  cents  per  cwt.  what 
will  be  the  first  cost  of  one  dozen  of  rolls,  each  weighing  5oz. 
allowing  the  bread  to  be  in  proportion  to  the  flour,  as  five  is 
co  four  ?  Ans.  6  cents. 

6.  If  a  merchant  bought  270  barrels  of  cider  for  780  dol- 
lars, and  paid  for  freight  37  dolls.  70  cents,  and  for  other 
charges  and  duties-  30  dolls.  60  cents ;  at  what  must  he  sell 
it  per  barrel  to  gain  143  dolls.  Ans.  3  dolls.  67^4T  cts. 

7.  If  half  a  ton  of  hay  was  equally  divided  among  80 
horses,  how  much  must  be  given  to  7  ?         Ans.  3qr.  14Z&. 

8.  Suppose  the  circumference  of  one  of  the  larger  wheels 
of  a  wagon  to  be  12  feet,  and  that  of  one  of  the  smaller 
wheels  9  feet  3  inches ;  in  how  many  miles  will  the  smaller 
wheel  make  1000  revolutions  more  than  the  larger? 

Ans.  7  m.  5  fur.  34  yds.  1  ft.  7  T\  in. 

9.  If  a  man  perform  a  journey  in  18  days,  when  the  days 
are  15  hours  long,  how  many  days  will  it  require  to  per- 
form the  same  journey  when  the  days  are  only  12  hours 
long?  Ans.  22^  days. 

10.  A  merchant  bought  a  piece  of  broadcloth  measuring 
42£  yds.  for  191  dolls.  25   cents;  15  yards  of  this   being 
damaged,  he  sells  it  at  two-thirds  of  its  cost ;  the  residue  he 
is  willing  to  sell  so  as  to  gain  1  doll,  per  yard  on  the  whole 
piece ;  at  what  rate  must  he  sell  the  remainder  ? 

Ans.  6  dolls.  86T4T  cents  per  yd. 

11.  If  60  yards  of  carpeting  will  cover  a  floor  that  is  30 
feet  long  and  18  broad,  what  is  the  width  of  the"  carpeting  ? 

Ans.  3  feet. 

12.  ff  a  piece  of -land  be  40   rods  in   length,  how  wide 
must  it  be  to  contain  4  arres  ?  Ans.  1 6  rods. 


64  SINGLE  RULE  OF  THREE. 

13.  Suppose  a  large  wheel,  in  mill  work,  to  contain  70 
cogs,  and  a  smaller  wheel,  working  in  it,  to  contain  52  cogs; 
in  how  many  revolutions  of  the  greater  wheel,  will  the  les- 
ser one  gain  100  revolutions?  Ans.  2'88£. 

14.  The  number  of  pulsations  in  a  healthy  person  is,  say 
70  in  a  minute,  and  the  velocity  of  sound  through  the  air  is 
found  to  be  1142  feet  in  a  second :  now  I  counted  20  pulsa- 
tions between  the  time  of  observing  a  flash  of  lightning  from 
a  thunder  cloud,  and  hearing  the  explosion  of  the  thunder; 
what  was  the  distance  of  the  cloud  ? 

Ans.  3  m.  5  fur.  145  yds.  2j  ft. 

15.  A  merchant  bought  5  pieces  of  cloth,  of  different 
qualities,  but  of  equal  lengths,  at  the  rate  of  5,  4,  3, 2,  and  1 
doll,  per  yd.  for  the  different  pieces ;  the  whole  came  to  532 
dolls.  50  cents ;  how  many  yards  did  each  piece  contain  ? 

Ans.  35£  yds. 

16.  What  principal  will  gain  as  much  in  1  month,  as  127 
dollars  would  gain  in  12  months?  Ans.  1524  dolls. 

17.  If  a  pair  of  steelyards  be  36  inches  in  length  to  the 
centre  of  motion,  the  pea  5  Ib.  and  the  draught  end  \  inch 
in  length,  what  weight,  will  they  draw?  Ans.  360  Ib. 

18.  Supposing  the  nbovp  steelyards  would  only  Hrow  00 
Jb.,  what  is  the  length  of  the  draught  end  ? 

Ans.  2  inches. 

19.  If  1  yard  of  cloth  cost  2  dolls.  71  cts.  1J  mills,  what 
will  67i  yards  come  to  at  the  same  rate? 

Ans.  183  dolls.  4  c.  6£  m. 

20.  If  a  man's  income  be  16s.  5d.  l-J^qr.  per  day,  what 
is  it  per  annum  ?  Ans.  300Z. 

-21.  How  many  pieces  of  wall  paper  that  is  3  qrs.  wide. 
and  11  yards  long,  will  it  require  to  paper  the  walls  of  a 
room  that  is  25  feet  long,  15  wide,  and  10-£  high,  allowing  a 
reduction  of  j\  for  doors  and  windows?  Ans.  lQ-fT. 

22.  The  length  of  a  wall  being  tried  by  a  measuring  line, 
appears  to  be  1287  feet  4  inches;  but  on  examination  the 
line  is  found  to  be  50  feet  10£  inches  in  length,  instead  of 
50  feet  its  supposed  length;   rrquirod   the  true  length  of  the 
wall?  Ans.  1309  feet  10}  J  inch. 

23.  If  a  dealer  in  liquors  use,  instead  of  a  gallon,  a  mea- 
sure which  is  deficient  by  half  a  pint,  what  will  be  the  true 
measure  of  100  of  these  false  gallons?         Ans.  93j  galls. 


DOUBLE  RULK  OF  THREK  DIRECT.  66 

SECTION  3. 

THE  DOUBLE  RULE  OF  THREE. 

THE  DOUBLE  RULE  OF  THREE,  or  as  it  is  often  called, 
Compound  Proportion,  is  used  for  solving  such  questions  as 
have  five  terms  given  to  find  a  sixth.  In  all  questions  be- 
longing to  this  rule,  the  three  first  terms  must  be  a  supposi- 
tion, the  two  last  a  demand. 

RULE  FOR  STATING. 

1.  Set  the  two  terms  of  the  supposition,  one  under  the 
other. 

2.  Place  the  term  of   the  same  kind   with  the  answer 
sought  in  the  second  place. 

3.  Set  the  terms  of  the  demand,  in  the  third  place,  ob- 
serving to  place  the  correspondent  terms  of  the  supposition 
and  demand  in  the  same  line.    Consider  the  upper  and  lower 
extremes,  with  the  middle  terms,  separately,  as  in  the  single 
rule  of  three ;  if  both  lines  are  direct,  then  the  question  will 
be  in  direct  proportion ;  but  if  either  lines  are  inverse,  then 
will  the  question  be  in  inverse  proportion. 

When  the  question  is  in  direct  proportion,  multiply  the 
product  of  the  two  last  terms  by  the  middle  term  for  a  divi- 
dend, and  multiply  the  two  first  terms  for  a  divisor;  the  quo- 
tient will  be  the  answer  in  the  same  name  with  the  middle 
term. 

But  if  the  proportion  be  inverse,  transpose  the  inverse 
terms  and  proceed  in  the  same  manner  as  in  direct  proportion. 

DIRECT  PROPORTION. 

EXAMPLE. 

1.  It'  6  men  in  8  days  earn  100  dollars,  how  much  will 
12  men  earn  in  24  days  ? 

dolls. 
6  men  )      ,     mn  5  12  men 


>t      \ 
f  ' 


8  days  f      '     J  )  24  days 

48  288 

100 


48)  28800  (600  dolls.  AM. 

288 

00 
F2 


06  DOUBLE  RULE  OF  THREE  INVERSE 

2.  If  10  bushels  of  oats  suffice  18  horses  for  20  days, 
how  many  bushels  will  serve  60  horses  36  days  ? 

Ans.  60  bushels. 

3.  If  56  pounds  of  bread  will  suffice  7  men  14  days,  how 
much  bread  will  serve  21  men  3  days  ?      Ans.  36  pounds. 

4.  If  8  students  spend  384  dollars  in  6  months,  how  much 
will  maintain  12  students  10  months?       Ans.  960  dollars. 

5.  If  20  hundred  weight  is  carried  50  miles  for  25  dol- 
lars, how  much  must  be  given  for  the  carriage  of  40  hun- 
dred weight  100  miles?  Ans.  100  dollars. 

6.  If  14  dollars  interest  is  gained  by  700  dollars  in  t> 
months,  what  will  be  the  interest  of  400  dollars  for  5  years '.' 

Ans.  80  dollars. 

7.  If  4  men  can  do  12  rods  cf  ditching  in  6  days,  how 
many  rods  may  be  done  by  8  men  in  24  days  ? 

Ans.  96  rods. 

INVERSE  PROPORTION. 

EXAMPLE. 

1.  If  4  dollars  pay  8  men  for  3  days,  how  many  days 
must  20  men  work  for  40  dollars  ? 

days. 

As  4  dolls.  (  q  J  40  dolls. 

8  men.  \  )  20  men. 

Here  the  lower  line  is  inverse,  which  transposed  will  stand 
thus :  <t 

days. 

As  4  dolls.  (          o  )    40  dolls. 

20  men.  {  )      8  men. 

80  320 

3 

80 )  960  ( 12  days.  Ans. 
80 

160 
160 

2.  If  4  men  are  paid  24  dollars  for  3  days  work,  how 
many  days  may  16  men  be  employed  for  3^4  dollars? 

Ans.  12  days. 

JJ.   If  4  men  arr  {-aid  '24  dollars   f<;r   3  days  work,  hou 
many  mon  may  !/o  rmployrfl  16  days  for  96  dollar? 

Ana.  3  men. 


DOUBLE  KULK  OF  THRICE.  (57 

4.  If  7  men  can  reap  84  acres  of  grain  in  12  days,  bow- 
many  men  can  reap  100  acres  in  5  days  ?     Ans.  20  men. 

5.  If  7  men  can  reap  84  acres  of  grain  in  12  days,  how 
many  days  will  it  require  20  men  to  reap  100  acres  ? 

Ans.  5  days. 

6.  If  40  cents  are  paid  for  the  carriage  of  200  pounds 
for  40  miles,  how  far  may  20200  pounds  be  carried  for  60 
dollars  60  cents  ?  Ans.  60  miles. 

7.  If  5  men  spend  200  dollars  in  22  weeks  and  6  days, 
now  long  will  300  dollars  support  12  men  ? 

Ans.  14  weeks  2  days. 

Promiscuous  Questions. 

1.  If  12  oxen  in  8  days  eat  10  acres  of  clover,  how  many 
acres  will  serve  24  oxen  48  days?  Ans.  120  acres. 

2.  A  person  having  engaged  to  remove  8000  weight  15 
miles  in  9  days  ;  with  18  horses,  in  6  days,  he  removed 
4500  weight ;  how  many  horses  will  be  necessary  to  re- 
move the  rest,  in  the  remaining  3  days  ?     Ans.  28  horses. 

3.  If  the  carriage  of  9  hogsheads  of  sugar,  each  weighing 
12  cwt.,  for  60  miles,  cost  100  dollars,  what  must  be  paid 
for  the  carriage  of  50  barrels  of  sugar,  each  weighing  2,5 
cwt.,  300  miles?  Ans.  578  dolls.  70  + cents. 

4.  If  1  pound  of  thread  make  3  yards  of  linen,  5  quar- 
ters wide,  how  many  pounds  of  thread  will  it  require   to 
make  a  piece  of  linen  45  yards  long  and  1  yard  wide  ? 

Ans.   12  Ib. 

5.  If  a  footman  travels  240  miles  in  12  days,  when  the 
days  are  12  hours  long ;  in  how  many  days  will  he  travel 
720  miles,  when  the  days  are  16  hours  long? 

Ans.  27  days. 

6.  A  perch  of  stone  measures  16^  feet  long,  1^  foot  broad, 
and  1  foot  high  ;  at  1  doll.  25  cts.  per  perch,. what  will  a 
pile  of  stone  *come  to,  which  measures  30  feet  long,  26  feet 
broad,  and  4£  feet  high  ?  .  Ans.  177  dolls.  27 -f  cts. 

7.  How  many  cords  are  there  in  a  pile  of  wood  200  feet 
long,  10  feet  high,  and  36  feet  broad ;  the  cord  measuring, 
according  to  law,  8  feet  in  length,  4  in  breadth,  and  4  in 
height  ?  Ans.  562J  cords, 

8.  If  3  pounds  of  cotton  make  10  yards  of  cloth,  6  qr. 
wide,  how  many  pounds  will  it  take  to  make  a  piece  100 
yards  long  and  3,  qr.  wide  ?  Am.  15  Ib 


68  PRACTICE 

9.  If  24  men  buiid  a  wail  200  feet  long,  8  feet  high,  and 
6  feet  thick,  in  80  days,  in  what  time  will  6  men  build  one 
20  ft.  long,  0  ft.  high,  and  4  ft.  thick  ?  Ans.  16  days. 

10.  If  a  family  of  9  persons  spend  450  dolls,  in  5  months, 
how  much  would  they  spend  in  8  months,  if  5  more  were 
added  to  the  family'?  Ans.  1120  dollars. 

11.  If  a  baker's  bill  for  a  family  of  8  persons  amounts  to 
11^:  dolls,  in  a  month,  when  flour  is  at  10  dolls,  per  barrel, 
what  will  it  amount  to  in  6  months  if  4  more  are  added  tc 
the  family,  and  flour  is  at  11  dollars  per  barrel  ? 

Ans.  Ill  dolls.  37£  cts. 

12.  If  a  cellar  which  is  22,5  feet  long,  17,3  feet  wide,  and 
10,25   deep,  is   dug  by  6  men  in  2,5  days,  working  12,3 
hours  each  day ;  how  many  days  of  8,2  hours  will  it  re- 
quire 9  men  to  dig  one  which  is  45  feet  long,  34,6  wide,  and 
12,3  feet  deep  ?  Ans.  12  days. 


PART  V. 
MERCANTILE  ARITHMETIC. 


SECTION  1. 

OF  PRACTICE. 

PRACTICE,  so  called  from  its  frequent  use  in  business,  is 
only  a  contraction  of  the  preceding  rules  of  proportion. 
By  it  a  compendious  way  is  given  of  finding  the  price  of 
any  given  quantity  of  goods  or  other  articles  of  trade,  when 
the  price  of  1  is  known. 

Case  1. 

When  the  price  consists  of  dollars,  cents,  and  mills. — 
Reduce  the  given  quantity  by  multiplication,  as  in  whole 
numbers,  and  point  off  from  the  right  of  the  product  for 
mills  and  cents,  according  to  the  rules  in  federal  money. 
Or,  multiply  by  the  dollars  only,  and  take  aliquot  or  frac- 
tional parts  for  the  cents  and  mills. 


PRACTICE 

TABLE. 

50  cents  is  £  of  a  dollar.          144  cents  is 
33J       .       £       .       .     ,          124       . 
25         .       i       .       .  11J       . 

20         .       i  .  10 

ief  .  ;t  5 

EXAMPLE. 

1.  W^hat  will  175  pounds  of  tea  come  to  at  1  dollar  30 
cents  and  5  mills  per  Ib.  ? 


T'O 


130,5 
175 

6525 
9135 
1305 

228375 


cts. 
Or,      25 
5 
5  m. 


i 
J 

TV 


175 
4375 

875 
875 

228375 


2. 
3. 

4. 
5. 
6. 

7. 
8. 


228  dolls.  87  cts.  5  mills. 
Z).  c.  m.  D.  c.  7 

250  yards  at  1,75  Ans.  437,50 


201  do.    4,20  884,20 

2210  do.    1,10  2431,00 

421  do.    2,41,5  1016,71,5 

625  do.     25  156,25 

8275  do.      4,4  864,10 

8275  do.       5  41,37,5 

Case  2. 

When  the  price  is  the  fractional  part  of  a  dollar,  or  cent, 
such  as  J  of  a  dollar,  f  of  a  cent,  multiply  the  quantity  by 
the  numerator,  and  divide  the  product  by  the  denominator  ; 
the  quotient  will  be  the  answer. 

EXAMPLE. 

1.  What  will  375  yards  of  muslin  cost  at  }  of  a  dollar 
per  yard  ? 

375  at  | 
3 


4 ) 1125 


cwt.  qr 
2.     4     1 
8.  12     2 
4.   14     2 


281,25    Ant. 
Ib.  D.  c.  m. 

14  of  sugar  at  £  of  a  doll,  per  Ib.  Ans.  122,50 
13  of  spice  at  f        do.        do.  942 

7  of  lead  at  J        do.      for  Mb.          285,42,5 


70  PRACTICE. 

Application. 

1.  Bought  6  hogsheads  of  tobacco,  each  weighing  12,5 
cwt.  at  f  of  a  doll,  per  pound  ;  what  did  it  cost  ? 

Ans.  3150  dolls. 

2.  A  gentleman  bought  a  vessel  of  60  tons  burden,  and 
gave  at  the  rate  of  2J  eagles  per  ton  ;  what  did  the  vessel 
cost  1  Ans.  1560  dolls. 

3.  A  carpenter  bought  12650  feet  of  boards  at  10 £  dol- 
lars per  thousand  ;  what  did  they  cost  him  ? 

Ans.  137  dolls.  56  cts.  8£  m. 

Case  3. 

When  the  price  and  quantity  given  are  of  several  denom- 
inations, multiply  the  price  by  the  integers,  or  whole  num- 
bers, and  take  aliquot  parts  for  the  rest. 

Table  of  a  hundred  weight. 


56  Ib.       is  of  a  Cwt. 

28  4 

16  .  . 


141b.       is      4  of  a  Cwt. 
•      A 


EXAMPLE. 


1.  Bought  Wcwt.  Iqr.  IQlb.  of  tobacco  at  12  dollars  44 
cents  per  hundred  weight ;  what  did  it  cost  ? 

12,44 
16 


Iqr.  is  1  4  I  19904 
1Mb.       I  4  I       311 

1777} 

2039274     Ans.  203  dolls.  92  cts.  7|  ms. 

2.  ITcwt.  3qr.  I9lb.  of  sugar  at  10  dollars  94  cents  per 
nundred  weight.  Ans.  196  dolls.  4cts. 

3.  5c*ot.  Iqr.  Qlb.  of  tobacco  at   13  dollars  41  cents  per 
hundred  weight.  Ans.  70  dolls.  40  cts.  2  m. 

4.  7&vt.  Qqr.  I9lb.  of  sugar  at  15  dolls.  5  mills  per  hun- 
dred weight.  Ana.  107  dolls.  58  cts. 


PRACTICE 


71 


Case  4. 

When  the  price  consists  of  pounds,  shillings,  pence,  and 
farthings. 

1.  Reduce  the  given  price  to  dollars  and  cents,  (see  re- 
duction of  money,  page  41)  and  then  proceed  according  to 
the  foregoing  cases.    Or, 

2.  Multiply  by  the  integers,  and  take  aliquot  parts  for  the 
remainder. 


of  a  shilling. 


TABLE. 

,?.    d. 

J. 

10            is 

i  of  a  pound. 

6 

68    — 

J 

4 

5           — 

3 

4 

f 

2 

34    — 

i 

26    — 

i 

2           — 

TV 

1      8    — 

Jy 

EXAMPLES. 

1.  What  will  4548  yards  come  to  at  Is.  fid.  per  yard  ? 
Is.  6d.=  18d.  —  20  cents=  }  of  a  dollar. 
cts. 
20  4548 


909,60  dollars.  Ans. 

Or,     6d.     |  i  |    4548  at  1  shilling,  will  be  the  same  num- 
2274         oer  of  shillings. 


2|0  )  682,2  shilling* 

£.341  2  =  909  dolls.  60  cents. 


473  yards  at  6 

397       do.  3 

\5§\lb.  of  coffee  at  1 
658/6.  of  tea  at  12 
745  yds.  of  cloth  at  16 
969*  do.  19 

3715  do.  9 


4. 
5. 
6. 
7. 
8. 
9.  4567 


d. 

£. 

s. 

d.   D.  c. 

8 

Arc*.  157 

13 

4=420,44$ 

4 

66 

3 

4  = 

8 

13 

5 

5r= 

394 

16 

0  — 

596 

0 

0= 

11 

964 

19 

a» 

4i 

1741 

8 

ij— 

do. 


19  114         4557 


9     «i  = 


72 


PRACTICE. 


Case  5. 

When  both  the  price  of  the  integer  and  the  quantity  are 
of  different  denominations,  multiply  the  price  by  the  inte- 
gers of  the  quantity,  and  take  parts  of  the  price  for  those 
of  the  integer. 


EXAMPLES. 


1.  What  will  45cu?i.  2qr.  14Z&.  of  sugar  come  to  at  37. 

7 s.  9d.  per  hundred  weight? 


5s.  is 
2 

6d.  is 
3 

t 

TV 
i 

i 

i 

i 

45     2  14 
379 

135 
11     5 
4  10 
1     2     6 
11     3 
1   13  104 
8     54 

2qr.  is 
1Mb. 

Or  thus, 


£. 

3 


d. 
9 
5X9=45 


16  18 


152     8     9 
2qr.  is  4        1   13  104 
14Z6.       i  8     54 


£.154  11     1  Ans. 


£.154  11     1 


2.  37  T.  llcwt.  2qr.  14Z6.  of  hemp  at  89Z.  65.  Sd.  per 
ton.  Ans.  3370Z.  13s.  2d. 

3.  3T.  I2cwt.  Sqr.  27lb.  of  sugar  at  bZ.  11s.  5d.  per  cwt. 

Ans.  6251.  11s.  Wd. 

4.  IT.  Icwt.  2qr.  2llb.  of  rice  at  3Z.  17s.  6d.  per  cwt. 

Ans.  5601.  13s.  3Jd. 

5.  476  acres  3  roods  28  perches  of  land  at  3Z.  7s.  lid. 
per  acre.  Ans.  1619Z.  11s.  l$d. 

6.  640  acres  2  roods  20  perches  at  10  dollars  55  cents 
per  acre.  Ans.  6758  dolls.  594  cts- 

7.  229  acres  3  roods  18  perches  at  18  dollars  50  cents 
per  acre.  Ans.  4252  dolls.  454  cts. 

9.  I2cwt.  Qqr.  lib.  at  6  dollars  34  cents  per  cwt. 

Ans.  76  dolls.  47  cts.  6  m. 

9.  llcwt.  3qr.  24JZ&.  at  14  dollars  per  cwt. 

Ans.  251  dolls.  56  cts.  24  m. 

10.  16  acres  3  roods  25  perches,  at  125  dollars  50  cents 
per  acre.  Ans.  2121  dolls.  73  cts.  3-f  m. 

11.  25ctrf.  3gr.  14Z/>.  at  3Z.  17*.  6d.  per  cwt. 

AIM.  1007.  5*.  3j<f. 


TARE  AND  TRET.  73 

SECTION  2. 

Allowance  on    the  weight  of  goods,  called 
Tare  and  Tret. 

Tare,  is  an  allowance  made  for  the  weight  of  the  barrel, 
box,  trunk,  &c.  in  which  goods  are  packed. 

Tret,  is  an  allowance  to  retailers,  for  waste  in  the  sales 
of  their  commodities. 

Gross,  is  the  whole  weight  of  the  .goods  with  the  barrel, 
box,  &c.  in  which  they  are  put  up. 

Neat,  is  the  weight  of  the  goods  after  all  allowances  are 
deducted. 

RULE. 

Find  the  amount  of  the  tare,  and  subtract  it  from  the  grosi 
weight,  the  remainder  will  be  the  neat. 

EXAMPLE. 

1.  What  is  the  neat  weight  of  a  hogshead  of  tobacco, 
weighing  gross  I2cwt.  3qr.  I2lb.  tare  14Z&.  per  cwt. 
Ib.  cwt.  qr.   ib. 

14  12     3     12  gross 

12  1     2     12  tare 


168=  the  tare  of  I2cwt.         Am.  11     1       0  neat 
10  8oz.  =  the  tare  of  3qr. 
1  Soz.~  the  tare  of  I2lb. 


180=  Irw*.  2$r.  12Z&. 

2.  What  will  3  barrels  of  sugar  come  to,  weighing  as  fol- 
<ows:  viz.  No.  1,  2cwt.  Iqr.  25/6.  No.  2,  2cwt.  2qr.  No.  3, 
2cwt.  2llb.  tare  2llb.  per  barrel  ;  at  12  dollars  50  cents  per 
cwt.?  An,s.  82  dolls.  47  cts.  7  m. 

3.  At  45  cents  per  pound,  wnat   wil  14  barrels  of  indigo 
come  to,  weighing  as  follows  : 

cwt.  qr.     Ib.  Ib. 

No.  1,—  3     3       2          Tare          29 
No.  2,  —  4      1      10  38 

\^  3,  —  4     o     ]9  32 

No.  4,  —  4     00  —  35 


Am.  760  dolls.  95  cts. 


74  INTEREST. 

4.  Bought  2  hogsheads  of  sugar,  weighing  as  follows 
viz.  No.  1,  llcict.  Iqr.  lllb.  tare  112Z6.  No.  2,  I2cwt.  2qr 
rare  74ZZ>.  at  16  dollars  80  cents  per  cwt.  neat;  for  which 
I  gave  18  barrels  of  flour  at  4  dollars  50  cents  per  bbl.  and 
I  %  ton  of  iron  at  120  dollars  per  ton;  what  was  the  balance 
still  due?  Ans.  112  dolls.  65  cts. 

5.  What  is  the  neat  weight  of  12  barrels  of  potash,  each 
weighing  ^cwt.  2qr.  26lb.  tare  I2lb.  per  cwt. ;  and  what 
will  it  come  to  at  9  dollars  per  cwt.  ? 

Ans.  SQcwt.  2qr.  23lb.  and  comes  to  456  dolls.  34|  cts. 

6.  Sold  a  hogshead  of  sugar,  weighing  6cwt.  gross,  tare 
100Z&.  tret  4Z&.  per  104,  for  82  dolls.  50  cents ;  what  was  it 
sold  for  per  pound?  Ans.  15  cents. 

7.  In  I20cwt.  3qr.  gross,  whole  tare  llllb.  tret  4Z6.  per 
104,  how  much  is  the  neat  weight  in  pounds,  and  what  the 
amount  at  73  cents  per  pound? 

Ans.  12833,6Z&.  and  comes  to  9368  dolls.  53  cts. 

8.  Bought  9   hogsheads  of  sugar,  each  weighing  6cwt. 
'2qr.  I2lb  gross ;  tare  lllb.  per  cwt.  what  is  the  neat  weight, 
and  wh?'  Joes  it  amount  to  at  16  dollars  per  cwt.  ? 

Ans.  5Qcwt.  Iqr.  22lb.  amounts  to  807  dolls.  14£  cts. 

9.  Sold  27  bags  of  coffee,  each  2cwt.  3qr.  lllb.  gross; 
ea re   l'3ib.  per  cwt.   tret  4Z&.   per   104;   what  is  the  neat 
weight,  and  what  will  it  co*ne  to  at  32  cents  per  pound  ? 

Ans.  66cwt   2qr.  lllb.  and  comes  to  2386  dolls.  88  cts. 


SECTION  3. 

OF  INTEREST. 

I  NT  FOREST  is  a  compensation  allowed  for  the  use  of  money, 
ibi  a  givim  time;  and  is  generally  throughout  the  United 
States  fixed  by  law  at  the  rate  of  6  dollars  for  every  100, 
;-K>;-  annum. 

1.  Thr  sum  of  w  <  .ney  at  interest,  is  called  the  Principal. 

^.  The  sum  per     -nt.  agreed  on,  is  called  the  Rate. 

3.  Tho  principal  id  interest  added  together,  is  called  the 
A/Hf*itn.t.  * 

Interest  is  firher.     nfflc  or  compound. 


SIMPLE  INTEREST.  75 

SIMPLE  INTEREST. 

SIMPLE  INTEREST  is  a  compensation  arising  from  the 
principal  only. 

Case  1. 

When  the  given  time  is  one  or  more  years,  and  the  prin- 
cipal dollars  only. 

RULE. 

Multiply  the  given  sum  by  the  rate  per  cent. ;  the  product 
will  be  the  interest  for  one  year  in  cents,  which  multiplied 
by  the  number  of  years,  will  be  the  answer  required. 

EXAMPLES. 

1.  What  is  the  interest  of  454  dollars  for  one  year,  at  6 
per  cent.  ? 

454 
6 


2724  cents.     Ans.  27  dolls.  24  cts. 
2.  Required  the  interest  of  the  same  sum  for  5  years,  at 
the  same  rate  ? 

454 
0 


2724 
5 

13620  cents.     Ans.  136  dolls.  20  cts. 
3.  Required  the  amount  of  the  same  sum  for  5  years,  at 
he  same  rate  ? 

454 
6 


2724 
5 

13620  interest. 
4.5400  principal. 


59020  amount.     Ans.  590  dofts.  20  cts. 
4.  What  is  the  interest  of  200  dollars  for  2  years,  at  6 
per  cent  ?  Ans.  24  dolls. 


76  SIMPLE  INTEREST. 

5.  What  is  the  interest  of  1260  dolls,  for  4  years,  at  7 
per  cent.  ?  Ans.  352  dolls.  80  cts. 

6.  What  is  the  amount  of  a  note  for  560  dollars  for  3 
years,  at  8  per  cent.  1  Ans.  694  dolls.  40  cts. 

7.  WThat  sum  must  be  given  to  discharge  a  bond  given  for 
4520  dollars,  on  which  there  is  6  years  interest  at  5  per 
cent.  ?  Ans.  5876  dolls. 

Note.  When  the  rate  per  cent,  contains  a  fraction,  such  as  £,  ^,  f 
the  principal  must  be  multiplied  by  the  fraction,  as  well  as  the  whole 
number  :  this  may  be  done  either  by  adding-  the  parts  of  \,  i,  &.c.  of 
the  principal  to  the  product  of  the  whole  number;  or  reduce  the  frac- 
tion to  a  decimal.  See  case  1,  in  reduction  of  decimals. 

8.  Wrhat  is  the  amount  of  400  dollars  for  2  years,  at  6| 
per  cent.  1  Ans.  452  dolls. 

9.  What  is  the  interest  of  4925  dollars  lor  y  years,  at  7^ 
per  cent.  ?  Ans.  3324  dolls.  37  cts.  5  m. 

10.  What  is  the  amount  of  2500  dollars  for  1  year,  at  7$ 
per  cent.  ?  Ans.  2693  dolls.  75  cts. 

Case  2. 

When  the  principal  is  dollars  and  cents,  or  dollars,  cents, 
and  mills,  and  the  time  years  only. 

RULE. 

Multiply  the  given  sum  by  the  rate  per  cent,  and  divide 
the  product  by  100  ;  or,  what  is  the  same,  point  off  two 
figures  on  the  right  of  the  product ;  .the  quotient,  or  remain- 
ing figures,  will  be  the  answer,  in  the  same  name  with  the 
lowest  denomination  in  the  principal. 

EXAMPLES. 

1.  What  is  the  interest  of  264  dollars  50  cents  for  1  year, 
at  6  per  cent.  ? 

264,50 
6 


cents  1587,00     Ans.   15  dolls.  87  cts. 
2.  What  is  the  interest  of  468  dollars  22  cents  and  5 
mills  for  1  year,  M  8  per  cent. 
468,225 


mills  37458,00     Ans.  37  d.»|ls.  45  cts.  8  m. 


SIMPLE  INTEREST.  ^     77 

3.   What  is  the  interest  of  364  dollars  50  cents  for  5  years, 
at  6  per  cent,  per  annum  ? 

36450  Or,    36450 

6  30= product  of  the  rate 

and  time. 

218700         cents  10935,00 
5 


cents  10935,00    Ans.  109  dollars  35  cents. 

4.  What  is  the  amount  of  a  note  for  1260  dollars  50 
cents  and  5  mills  for  3  years,  at  7^  per  cent,  per  annum? 

Ans.  1544  dolls.  11  cts.  8-f  ms. 

5.  What  sum  will  discharge  a  bond  given  for  630  dollars 
50  cents,  on  which  there  is  5  years  interest  at  8  per  cent, 
per  annum  ?  Ans.  882  dolls.  70  cts. 

6.  What  is  the  difference  between  the  interest  of  1274 
dollars  64  cents  6  mills  for  3  years,  at  7J  per  cent,  per  an- 
num, and  the  interest  of  3462  dollars  84  cents,  for  4  years, 
at  3^:  per  cent,  per  annum  ? 

Ans.  The  lattep  is  163  dolls.  37  c.  3,85  m.  the  greater. 

7.  A  gave  B  his  bond  for  3422  dolls.  25  cents,  to  be  paid 
in  the  following  manner,  viz.  one-third  at  the  end  of  one 
year,  one-third  at  the  end  of  two  years,  and  the  remainder 
at  the  end  of  three  years,  with  interest  from  the  date,  at  6 
per  cent,  per  annum ;  what  will  be  the  annual  payments, 
and  what  the  whole  amount  ? 

Ans.  1st  payment  1209  dolls.  19,5  cts.  2d,  1277  dolls.  64 
cts.  3d,  f346  dolls.  8,5  c. ;  whole  amount  3832  d.  92  c. 

Case  3.     . 

When  the  principal  is  dollars,  cents,  &c.  and  the  time  is 
years  and  months,  or  months  only. 

RULE. 

Multiply  by  half  the  number  of  months  in  the  given  time, 
when  the  rate  is  6  per  cent,  per  annum :  but  if  the  rate  per 
cent,  be  more  or  less  than  6  per  cent.,  multiply  the  given 
number  of  months  by  the  rate,  and  divide  the  product  by 
12  ;  the  quotient  will  be  the  rate  for  the  time;  the  principal 
multiplied  by  this  rate,  will  give  the  interest  required. 
G2 


78        .,  SIMPLE  INTEREST 

EXAMPLES. 

1.  What  is  the  interest  of  650  dollars  for  8  months,  at  6 
per  cent,  per  annum  ? 

650 

4  half  the  months 


cents  2500     Ans.  26  dollars. 

2.  What  13  the  interest  of  860  dollars  for  1  year  and  6 
months,  at  6  per  cent,  per  annum  ? 
860 

9  half  the  months 


cents  7740     Ans.  77  dolls.  40  cents. 
3.  What  is  the  interest  of  420  dollars  for  9  months,  at  8 
per  cent,  per  annum  ? 

9  months  420 

8  per  cent.  6 


12  )  72  2520   Ans.  25  dolls.  20c. 

6  the  rate  for  the  time. 

4.  What  is  the  amount  of  a  note  for  724  dollars,  with  18 
months  interest  due  thereon,  at  4  per  cent,  per  annum  ? 

Ans.  767  dolls.  44  cts. 

5.  What  is  the  interest'  of  240  dollars  for  15  months,  at 
7  2  per  cent,  per  annum  ?  A.ns.  22  dolls.  50  cts. 

6.  What  is  the  interest  of  1260  dollars  for  4  months,  at 
G£  per  cent,  per  annum  ?  Ans.  27  dolls.  30  cts. 

Case  4. 

When  the  principal  is  dollars,  cents,  &c.  and  the  time  is 
months  and  days,  or  days  only. 

RULE. 

Find  the  interest  for  the  given  months  by  the  last  case, 
find  take  aliquot  parts  for  the  days. 

Note.     In  calculation  of  interest,  30  days  make  a  month. 

Or,  multiply  the  given  sum  when  the  rate  is  6  per  cent. 
i)\  the  number  of  days,  and  divide  the  product  by  60  ;  the 
<]ii<>;ient  is  the  interest  required. 

Note.  Though  both  the  foregoing  methods  are  considered  sufficiently 
exart  for  common  business,  by  merchants  and  jiccomptants  generally, 


SIMPLK  INTKRKST.  7<J 

yet  as  this  is  only  allowing  360  days  in  the  year,  and  not  365,  the  true 
time ;  if,  therefore,  the  principal  is  large,  on  which  interest  is  due,  and 
greater  exactness  is  required,  then  find  the  interest  of  the  given  sum 
tor  1  year,  and  proceed  according  to  the  single  rule  of  three.  As  365 
days  :  to  the  interest  for  one  year  ::  the  given  number  of  days  :  the  an 
swer.  Or  by  the  double  rule  of  three,  find  the  fixed  divisors,  which  fbi 
5  per  cent,  is  7300,  fer  6  per  cent,  is  6083,  for  7  per  cent.  ^214 ;  multi- 
ply the  principal  by  the  days,  and  divide  by  these  divisors  according  to 
the  rate  per  cent,  required. 

EXAMPLES. 

1.  What  is  the  interest  of  260  dollars  for  5  months  aid 
20  days,  at  6  per  cent,  per  annum  ? 

260 
2 

\  month      I  i  |5,2G     interest  ibr  4  months 
15  days        [  i  UfSO     interest  for  1      do- 
5  j |4  I    65     interest  for  15  days 

2 1,6  interest  for  5     do- 

7,36,6  Ans.  7  dolls.  36  cts.  6  m. 

2.  What  is  the  interest  of  450  dollars  for  36  days,  at  ti 
per  cent,  per  annum  ? 

450  6,0 )  1620,0 

36 

2,70  Ans.  2  dolls.  70  cts. 
2700 
1350 

16200 

3.  What  is  the  interest  of  564  dollars  for  44  days,  at  6 
per  cent,  per  annum  ? 

564 
6 

days       days 

Ai         365    :    3384    ::    44 
44 


13536 
13536 

365 )  148896  ( 4,079,    Ans.  -i  dolls.  07  cts.  9  -f  m. 
1460 

2896 
2555 

3410 
3285 

125 


80  SIMPLE  INTEREST. 

4.   What  is  the  interest  of  960  dollars  for  70  days,  at  6 
per  cent,  per  annum  ? 
960 

"  70 


6083  )  67200  ( 1 1  +  dolls.    Ans. 
6083 

6370 
6083 

287 

5.  What  is  the  interest  of  12000  dollars  for  40  days,  at 
7  per  cent,  per  annum  1  Ans.  92  dolls.  6  cts. 

6.  What  is  the  interest  of  8400  dollars  for  20  days,  at  5 
,,er  cent,  per  annum  ?  Ans.  23  dolls. 

D.  c.  days  D.c.m. 

7.  517,90  for  84  at  6  per  cent,  per  annun*.    Ans.  7,15,1 

e.  73,41  27 33 

9.  225,24  40  ....  1,48,1 

10.  1200,00  80  -  15,78,1 

11.  ^62,19  254 123,68,8 

12.  1733,97  102  -  29,07,5 

A  TABLE, 

knowing  the  number  of  day  s+ from  any  day  in  any  mojtth,  to  the  same 
day  in  any  other  month  through  the  year. 


From   jJa.|Fb.|Mr.|Ap.|Ma.  Ju.  |Jly.|Au.|  Se.|Oc.|No.|De. 


ToJan.|365|334|306|275|245|214|184|153|122|   92|   61|   31 


Feb.     |   31j365|337|306[276|245|215|184|153|123|   92|   62 


59|  28|365|334|304|273|243|212!lftl|151|120|   90, 


April    |   90|  59|   31|365|335|304|274',243|212|182|151|121 


May     |120|  89|  61|   30|365J335|304|273|242|212|181|151 


|151|120|'  92 1  61|   31|365J885|304!273|243|212|182 


July     |181|150|122|   91|   61|   30|365|334|303|273|242|212 


Au?ust|2121181|153|122|   92 1   61 1   31|365|334|304|273(243 


sept.    |243|212|184|153|123|   92|  62|   31|365|335|304|274 


Oct.      |273|242|214|183|153|122|   92|   61 1   30|365|334|3Q4 


|304|273|245|214|184|153|123|   92|  61 1   31;365|335 


|334|303|275|244|214|183|153|122|   91 1   61|  30|365 


Suppose  the  number  of  days  between  the  10th  of  April 
and   l()th  of  Ortohpr  wpre   rerjuirorl ;  under  the  column  of 


SlMl'LK  1NTKRKST.  8) 

April  at  the  top  of  the  table,  look  for  October,  and  you  find 
=188,  the  number  required. 

Ii  the  days  in  the  given  months  be  different,  their  differ- 
ence must  be  added  or  subtracted,  to  or  from  the  tabular 
number.  Thus,  from  the  10th  of  April  to  the  20th  of  Oc- 
tober, is  183  +  10  =  198  days.  And  from  the  20th  of  April 
to  the  10th  of  October,  183—10=173  days. 

If  the  time  exceed  a  year,  365  days  must  be  added  for 
each  year. 

Case  5. 

When  the  amount,  rate,  and  time  are  given  to  find  the 
principal. 

RULE. 

As  the  amount  of  100  dollars,  at  the  rate  and  time  given, 
is  to  100  dollars,  so  is  the  amount  given  to  the  principal  re- 
•quired. 

EXAMPLE. 

1.  What  principal  being  put  to  interest  for  9  years,  at  5 
per  cent,  per  annum,  will  amount  to  725  dollars  ? 
9 


45 
100 

As      145      :      100     ::     725     :     500  Ans, 

2.  What  principal  being  put  to  interest  for  12  years,  at 
ft  per  cent,  per  annum,  will  amount  to  2752  dollars? 

Ans.  1600  dolls. 

3.  Received  728  dollars  as  payment  in   full   for  a  note 
with  5  years  interest  thereon,  at.  6  per  cent,  per  annum  ;  for 
how  much  was  the  note  given?  Ans.  560  dolls. 

4.  What  sum  put  to  interest  for  4  years,  at  7 5  per  cent. 
per  annum,  will  amount  to  1 638  dollars  ? 

Am.  1260  dolls. 

5.  Received  2000  dollars  as  payment  in  full  for  a  bond, 
with  5  years  interest  thereon,  at  5|  per  cent,  per  annum  ; 
what  principal  did  the  bond  contain  ? 

Ans.  1553  dolls.  39  cts.  8TJm. 


82  SIMPLE  INTEREST 

Case  6. 

When  the  amount,  time,  and  principal  are  given  to  find 
the  rat3. 

RULE. 

1.  As  the  principal  is  to  the  interest  for  the  whole  time 
so  is  100  dollars  to  its  interest  for  the  same  time. 

2.  Divide  the  interest  so  found  by  the  time,  and  the  quo- 
tient  will  give  the  rate  per  cent. 

EXAMPLE. 

1.  At  what  rate  of  interest  per  cent,  will  500  dollars 
amount  to  725  dollars  in  9  years  ? 

725 
500 

D.  D.  D.          D. 

225    As     500     :     225     ::      100     :     45 
9)45 

5  per  cent.    Ans. 

2.  Paid  858  dollars  in  full  for  a  note  given  for  650  dollars, 
with  4  years  interest  due  thereon;  what  was  the  rate  per 
cent,  per  annum  charged  on  said  note  ?      Ans.  S  per  cent. 

3.  At  what  rate  per  cent,  will  1600  dollars  amount  to  275*2 
dollars  in  12  years?  Ans.  6  per  cent. 

4.  At  what  rate  per  cent,  will  640  dollars  amount  to  860 
dolls.  80  cents  in  6  years?  Ans.  5f  per  cent. 

5.  At  what  rate  per  cent,  will   12000  dollars  amount  to 
50100  dollars  in  15  years?  Ans.  4£  percent. 

Case  7. 

When  the  principal,  amount,  and  rate  are  given  to  find 
the  time. 

RULE. 

Find  the  interest  of  the  principal  for  one  year.  And  thens 
as  the  interest  for  1  year,  is  to  1  year,  so  is  the  whole  inter- 
est to  the  time  required. 

KX  AMPLE. 

1.  In  what  time  will  500  dollars  amount  to  725  dollars 
at  5  per  cent,  per  annum? 
500 
5 

D.         Y.          D.          Y. 


!, acres!  for  1  year    25,00    As     25     :     1      ::     22,)     :     <)  Ans. 
Amount    725 
Principal  500 

Whole  interest     22o 


Sl^lPLK  INTKRKST.  83 

2.  »ln  what  time  will  050  dollars  amount  to  910  dollars, 
at  S  per  cent,  per  annum  1  Ans.  5  years. 

3.  In  what  time  will  1600  dollars  amount  to  2080  dollars 
at  6  per  cent,  per  annum  ?  Ans.  5  years. 

Case  8. 

When  the  principal  is  in  English  money,  viz.  pounds-, 
shillings,  and  pence,  and  the  interest  required  either  in  Fed- 
eral or  English  money. 

RULE. 

Reduce  the  English  to  Federal  money,  and  find  the  in 
terest  by  the~preceding  rules. 

EXAMPLE. 

1.  What  is  the  interest  in  Federal  money,  of  325Z.  105. 
English  money,  for  5  years,  at  6  per  cent,  per  annum  ? 
£.    s         D.    c. 
325  10=1445,22 

30=the  time  multiplied  by  the  rate. 


4335660  Ans.  433  dolls.  56  cts.  6m 

2.  What  is  the  amount  of  a  note  for  640/.  3s.  6d.  with  3 
years  interest  due  thereon,  at  5  per  cent,  per  annum,  in  Fed- 
eral money?  Ans.  3268  dolls.  73  cts.  3|i  m. 

3.  What  is  the  interest  of  1374Z.  ls.-9d.  for  U  ycar:.  aJ 
5f  per  cent,  per  annum?  Ans.   1157.  185.  Or/. 

Case  9. 

Computing  interest  on  bonds,  notes,  &c.  on  which  differ- 
ent payments  have  been  made. 

RULE  I. 

Find  the  interest  of  the  principal  from  the  time  the  inter- 
est first  commenced,  to  the  time  of  the  first  payment  made  ; 
add  that  interest  to  the  principal,  and  subtract  from  the 
amount  the  payment  made  ;  the  remainder  forms  a  new 
principal ;  on  which  proceed  in  the  same  manner,  till  all  the 
payments  are  brought  in. 

Note  1.  When  a  payment  alone,  or  in  conjunction  with  any  pre- 
ceding- payment,  is. less  than  the  interest  due  at  the  time,  then  no  cal- 
culation must  be  made ;  but  these  lesser  payments  added  to  the  next 

2.  By  this  rule,  no  part  of  the  interest  ever  forms  a  part  of  the  prin- 
cipal carrying  interest,  the  payments  being  first  applied  to  discharge 
the  interest. 


84  SIMPLE  INTEREST. 

EXAMPLES.  * 

1.  A  has  B's  note  for  1000  dollars,  dated  1st  January 
1816,  payable  in  18  months,  with  interest  from  the  date,  at 
6  per  cent,  per  annum.  On  which  the  following  payments 
are  endorsed,  viz. 

1816.  July  1.  Rec'ci  on  the  within  note  230  dollars 

1817.  Jan.  1.  Rec'd  -     300 
March        1.  Rec'd         ...         4 
April          1.  Rec'd                            -     250 

What  was  the  balance  due  on  the  1st  of  July,  1817,  when 
the  whole  note  is  payable  ? 

D.     c. 

Principal  at  interest  from  January  1,  1816,        1000  00 
1816,  July  1.  Interest  (6  months)      -  30 

1030 
Paid  same  date 230 


Remainder  for  a  new  principal  -         800 

1817,  January  1.  Interest  (6  months)  '24 

824 
Paid  same  date         .....         300 


Remainder  for  a  new  principal  -         ^         524 

March  1 .  Paid  4  dollars  less  than  the  interest 

and  not  to  be  calculated. 
April  1.  Interest  (3  months)      -  7  86 

531   86 
Paid  same  date  250 +  4=          -         .      »   .         254 


277  86 
July  1.  Interest  (3  months)        -  4  17 

Balance  due  Ans.  282  03 

RULE  II. 

Multiply  the  principal  by  the  number  of  days,  till  the  first 
payment  is  made ;  the  remaining  principal  by  the  number 
of  days,  between  the  first  and  second  payment,  &c.  till  aU 
the  payments  are  made  ;  divide  the  whole  amount  by  60  ; 
the  quotient  will  give  the  interest  required. 

Note.  By  this  method  the  interest  is  generally  calculated  among 
merchants. 


INTEREST.  85 

1.  A  bond  was  given  by  B  to  C,  for  2400  dollars,  payable 
in  2  years,  with  interest  from  the  date.  Dated  July  1,  1815. 
On  this  bond  the  following  payments  are  eridgrsed  ;  viz. 
May  1,  1316,  900  dollars  ;  October  1,  1816,  450  dollars  , 
January  1,  1817,  620  dollars.  Required  the  amount  due  on 
the  1st  of  May,  1817? 

1815.  July  1.     Principal  2400  dollars. 

1816.  May  1.  2400  multiplied  by  304     is     729600 

Paid  900 


Oct.  1.  1500       ....     153     .     229500 

Paid  450 


1917.  Jan.  1.  1050       .     .-.     .       92     .       96600 

Paid  620 


May  1.  430       ....     120     .       51600 

Interest  184  55 

.        Divide  by  6,0  )  110730,0 

Balance  614  55    Ans. 

•  Interest     184,55 

2.  A  note  was  given  by  A  to  B,  for  1800  dollars,  dated 
1st  January,  1820,  with  interest  from  the  date.  On  which 
the  following  payments  are  endorsed,  viz.  April  1,  1821, 
700  dollars;  January  1,  1822,  400  dollars;  July  1,  1822, 
500  dollars.  Required  the  amount  due  on  the  1st  of  Janu- 
ary, 1823  ?  Ans.  414  dolls.  16  cts.  6  m. 


COMPOUND  INTEREST 

Is  a  compensation  allowed  not  only  for  the  principal,  but 
also  for  the  interest  as  it  becomes  due. 

RULE. 

Add  the  simple  interest  of  the  given  sum  for  one  year  to 
the  principal.     This  amount  forms  a  new  principal  for  the 
second  year,  and  so  on  for  any  number  of  years  required. 
Subtract  the  first  principal  from  the  last  amount,  the  re- 
mainder will  be  the  compound  interest  required. 

H 


COMPOUND  INTEREST. 


EXAMPLE. 

1.   What  is  the  compound  interest  of  500  dollars  for  3 
years,  at  6  per  cent,  per  annum  I 
dolls. 

500  1st  principal 
6 

30,00  interest 
500 


530,00  2d  principal 
6 


31,80,00 
530 

561,80  3d  principal 
6 


33,70,80 
561,80 

595,50,8  last  amount 
500 

95,50,8        Ans.  95  dolls.  50  cents,  8  mills. 

Compound  Interest  may  be  more  expeditiously  calcufctted  by  the  follow- 
ing Table,  in  which  the  amount  of  one  dollar  for  any  number  of  years 
under  30  is  shown,  at  the  rates  of  5  and  6  per  cent,  per  annum,  com- 
pound interest. 


Years 

5  Ra 

tes   6 

Years 

5  Ri 

ites   6 

1 

1.05000 

1.06000 

16 

2.18287 

2.54035 

2 

1.10250 

1.12360 

17 

229201 

2.69277 

3 

1.15762 

1.19101 

18 

2.40662 

2.85434 

4 

1.21550 

1.26247 

19 

2.52695 

3.02559 

5 

1.27628 

1.33822 

20 

2.65329 

3.207  1.5  , 

6 

1.34009 

1.41852 

21 

2.78596 

3.39956 

7 

.40710 

1.50363 

22 

2.92526 

3.60353 

8 

.47745 

1.59384 

23 

3.07152 

3.81i>75 

9 

.55132 

1.68948 

33 

3.22510 

4.04898 

10 

.62889 

1.79064 

25 

3.38635 

4.29187 

11 

.71034 

1.89829 

26 

3.55567 

l.r,ii)38 

12 

.79585 

2.01219 

27 

.T73345 

1234 

13 

1.88565 

2.l32;j^ 

28 

3.920  J  3 

5.11168 

14 

1.97993 

2.26090 

29 

4.11613 

5.41838 

15 

2.07892 

2.39655 

30 

4.32194 

5.74349 

To  find  the  compound  interest  of  any  sum  by  this  labic, 
multiply  the  figures  opposite  the  number  of  years,  under  the 
rate  percent,  b  r'n  prim-ipal  ;  the  product  will  be 

tbt:   amount    required  ;   from   this   subtract  the  principal,  tht. 
remainder  will  b''  th< 


INSURANCE  COMMISSION.  AND  BROKAGK.  87 

EXAMPLE. 

2.  What  is  the  compound  interest  of  1000  crbllars  for  W 
years,  at  6  per  cent,  per  annum  ? 

1,59384          the  tabular  number  for  the  time 
1,000  the  principal 


1593,84000 
1000 

593,84         the  interest.  Ans.  593  dolls.  84  cts. 

3.  What  is  the  amount  of  1500  dollars  for  5  years,  at  5 
pej*  cent,  per  annum?  Ans.  1914  dolls.  42  cts. 

4.  What  is  the  compound  interest  of  4500  dollars  for  16 
years,  at  6  per  cent,  per  annum  ? 

Ans.  6931  dolls.  57  cts.  5  m. 

5.  A  has  B's  note  for  650  dollars,  payable  at  the  end  of 
20  years,  at  6  per  cent,   per  annum,  compound  interest ; 
what  sum  will  it  require  to  discharge  the  note,  at  the  expira- 
tion of  the  given  time  ? 

Ans.  2084  dolls.  63  cts.  4  m. 

6.  A  father  left  a  legacy  of  8000  dollars  at  compound  in- 
terest, 6  per  cent,  per  annum,  to  be  equally  divided  among 
his  three  sons,  when  the  youngest,  who  was  4  years  old, 
should  arrive  at  the  age  of  21 ;  what  will   be  each  one's 
share?  Ans.  7180  dolls.  72  cts.  each  share. 


SECTION  4. 

Insurance,  Commission,  and  Brokage. 

INSURANCE  is  a  premium  given  for  insuring  the  owners  of 
property  against  the  dangers  and  losses  to  which  it  is  liable, 
or  indemnifying  for  its  loss. 

The  instrument  of  agreement  by  which  this  indemnity  is 
secured  is  termed  the  policy  of  insurance. 

Commission  is  a  compensation  allowed  to  merchants  and 
others  for  buying,  selling,  and  transporting  goods,  wares, 


88  INSURANCE  A\V>  COMMISSION. 

Brokage  is  an  allowance  given  to  brokers  for  exchanging 
money,  buying  and  selling  stock,  &c. 

The  method  of  operation  in  all  these  is  the  same  as  in 
simple  interest, 

EXAMPLES, 

INSURANCE. 

1.  What  is  the  premium  of  insuring  1260  dollars,  at  5 
per  cent  ? 

1260 
5 


63,00     Ans.  63  dolls. 

D.    c. 

2.  1650  dollars  at  15£  per  cent.         -         Ans.  255  75 

3.  4500  25         -          -  -        1125  00 

4.  What  sum  must  a  policy  be  taken  out  for,  to  cover 
900  dollars,  when  the  premium  is  10  per  cent? 

100  policy 
10  premium 

90  sum  covered 
As     90     :     100     ::     900     :     1000  dolls.   Ans. 

5.  What  sum  will  it  require  to  cover  a  policy  of  insurance 
for  4500  dolls,  at  25  per  cent  1  Am.  6000  dolls. 

6.  What  sum  witl.it  require  to  cover  a  policy  of  insurance 
for  560  dollars,  at  9  per  cent7         Ans.  615  dolls,  38$  cts, 

COMMISSION. 

1.  What  is  the  commission  on  850  dolls,  at  5  per  cent. 
850 
5 


42,50     Ans.  42  dolls.  50  cts. 

2.  What  is  the  commission  on  1260  dollars,  at  6  per 
rent?                                                        Ans.  75  dolls.  60  cts. 

D.  c. 

3.  2550  dollars  at  4  per  cent.      -     -     Ans.  102  00 

4.  26342  H 790  26 

5.  6422  } 48  10  J 


BROKAGE— Bl'YINU  AND  SELLING  STOCKS.  89 

6.  A  commission  merchant  receives  1260  dollars  to  fill 
an  order,   from  which  he  is  instructed  to  deduct  his  own 
commission  of  5  per  cent,  how  much  will  remain  to  satisfy 
the  order? 

100 

5  per  cent. 

As     105     :     100     ::     1260     :     1200  dolls.  Ans. 

7.  A  commission   merchant  has   received    4120  dollars 
\vith  instructions  to  vest  it  in  salt  at  8  dollars  per  harrel ;  de- 
ducting from  it  his  commission  of  3  per  cent,  how  many 
barrels  of  salt  can  he  purchase  ?  Ans.  500  barrels. 

BROKAGE. 

1.  What  is  the  brokage  on  1000  dollars,  at  lj  per  cent? 
1000 
H 


1000 
500 

15,00     Ans.  15  dolls. 

2.  What  is  the  brokage  on  1625  dollars  50  cents,  at  3^ 
per  cent?  Ans.  54  dolls.  18  cts. 

3.  1868  dollars  at  2£  per  cent.          Ans.  46  dolls.  70  cts. 

4.  560  6  38  60 


SECTION  5. 

BUYING  AND  SELLING  STOCKS. 

Stock  is  a  fund  vested  by  government,  or  individuals  in  a 
corporate  capacity,  in  banks,  turnpike  roads,  bridges,  &c. 
the  value  of  which  is  subject  to  rise  and  fall. 

RULE. 

Multiply  the  given  sum  by  the  rate  per  cent,  and  divide 
the  product  by  100 

H2 


90  REBATE  OR  DISCOUNT. 

EXAMPLES. 

1.  What  is  the  amount  of  1650  dollars,  United  States 
bank  stock,  at  125  per  cent,  or  25  per  cent,  above  par? 
1650 
125 


6250 
3300 
1650 

1,00  )  2062,50         Ans.  2062  dolls.  50  cts. 
Or  thus,  25  is  i)1650 

412  50 

2062  50  AM. 

D.  D.     c. 

1500  bank  stock  at  110  per  cent.    Ans.  1650  00 

1686           128  .   2158  08 

4.  25000           108  -    -  27000  00 

5.  1260            90  -    -   1134  00 

6.  9254            84  -   7773  36 

7.  1518            83|  1271  32$ 


SECTION  6. 
REBATE  OR  DISCOUNT, 

Is  a  reduction  made  for  the  payment  of  money  before  il 
tecomes  due.  It  is  estimated  in  such  a  manner,  as  that 
the  ready  payment,  if  put  to  interest  at  the  same  rate  and 
time,  vvould  amount  to  the  first  sum.  Thus,  6  dollars  is  the 
discount  on  106  dollars  for  12  months,  at  6  per  cent,  leaving 
100  dollars  the  ready  payment,  which,  if  put  to  interest  for 
the  same  rate  and  time,  would  regain  the  6  dollars  discount. 

RULE. 

As  100  dollars  and  the  interest  for  the  given  time,  is  to 
100  dollars,  so  is  the  given  sum  to  its  present  worth. — Sub 
tract  the  present  worth  from  tho  given  sum,  and  the  remain 
dor  is  the  discount. 


REBATE  OR  DISCOUNT.  91 

^  EXAMPLE. 

1.  What  s  the  discount  of  1696  dollars,  due  12  months 
hence,  at  6  per  cent,  per  annum  ? 

As    106     :     100     ::     1696     :     1600 
1600 


96  dolls.  Ans. 

2.  What  is  the  present  worth  of  2464  dollars,  due  1  year 
and  6  months  hence,  discounting  at  the  rate  of  8  per  cent. 
nor  annum]  Ans.  2200  dolls. 

3.  A  has  B's  note  for  1857  dollars  50  cents,  payable  8 
months  after  date ;  what  is  the  present  worth  of  said  note, 
discounting  at  the  rate  of  5J  per  cent,  per  annum? 

Ans.  1791  dolls.  80  cts. 

4.  What  reduction  must  be  made  for  prompt  payment  of 
a  note  for  650  dollars,  due  2   years  hence,  7  per  cent,  per 
annum  being  allowed  for  discount  ? 

AILS.  79  dolls.  83  +  cts. 

5.  What  is  the  present  worth  of  5150  dollars,  due  in  4^ 
months,  discounting  at  the  rate  of  8  per  cent,  per  annum, 
and  allowing  1  per  cent,  for  prompt  payment  ? 

Ans.  4950  dolls. 

Note.  Discount  and  interest  are  often  supposed  to  be  one  and  the 
same  thing;  and  in  business,  the  interest  for  the  time  is  frequently 
taken  for  the  discount,  and  it  is  presumed  neither  party  sustains  any 
loss.  This  however  is  not  true,  for  the  interest  of  100  dollars  for  12 
months,  at  6  per  cent,  is  6  dollars,  whereas  the  discount  for  the  same 
sum,  at  the  same  rate  and  ti'ne,  is  only  5  dollars  66  cents,  making  a 
difference  of  34  cents  for  every  100  dollars  for  1  year  at  6  per  cent. — 
The  following1  examples  will  show  the  difference. 

EXAMPLES. 

1.  What    is   the    discount   of   1272  dollars,   due    in   12 
months,  discounting  at  6  per  cent,  per  annum  ? 

.As    106     :     100     ::     1272     :     1200 

discount  72  dolls. 

2.  What  is  the  interest  of  the  same  sum,  for  the  same 
time  and  rate  ? 

1272  D.     c. 
6        .                       Interest     76     32 
Discount  72 


76,32  interest, 


Difference          4     32 


92  BANK  DISCOUNT. 

8.  What  is  the  difference  between  the  interest  and  dis- 
count on  7280  dollars,  for  18  months,  at  8  per  cent,  per  an- 
num? Ans.  93  dolls.  60  cts.  difference. 

Note.  But  when  discount  is  made  for  present  payment,  without  re- 
gard to  time,  the  interest  of  the  sum  as  calculated  for  a  year,  is  the 
discount. 

EXAMPLE. 

1.  How  much  is  the  discount  of  260  dollars  at  5  per 
cent? 

260 
5 


13,00     Ans.  13  dollars. 

2.  What  is  the  discount  on  1650  dollars,  at  3  per  cent! 

Ans.  49  dolls.  50  cte. 

3.  What  sum  will  discharge  a  bond  for  2464  dollars,  on 
which  a  discount  of  8  per  cent,  is  given  ? 

Ans.  2266  dolls.  88  cts. 


SECTION  7. 

BANK  DISCOUNT. 

BANK  discount  is  the  interest  which  banks  receive  for  the 
use  of  money  loaned  by  them  for  short  periods.  And  as 
banks  from  long  established  custom,  give  three  days  over 
and  above  the  time  limited  by  the  words  of  the  note,  called 
days  of  grace  ;  and  as  the  day  of  the  date,  and  the  day  of 
payment  are  both  calculated,  which  makes  the  time  4  days 
longer  than  expressed  in  the  note,  so  interest  must  be  calcu 
lated  on  these  days  in  addition  to  the  regular  interest  on  the 
given  sum,  for  the  specified  time. 

RULE. 

Add  4  to  the  number  of  days  specified  in  the  note,  multi- 
ply the  given  sum  by  this  number,  and  divide  the  product  by 
60.  Or, 

Multiply  the  given  sum  by  half  t)ie  number  of  days,  and 
divide  bv  30. 


EQUATION  OF  PAYMENTS.  f         98 

Note.  When  the  cents  in  the  given  sum  are  less  than  50,  the  bank 
loses  the  interest  on  them,  but  when  they  are  more  than  50  they 
charge  interest  for  one  dollar. 


EXAMPLES. 


1.  Required  the  discount  of  1500  dollars  for  60  days. 
1500  Or,    1500 

64  32= half  the  days 


6000  3000 

9000  4500 


6,0)9600,0  3,0)4800,0 


16,00    Ans.  16  dolls.  16,00 

2.  What  is  the  discount  of  250  dollars  for  30  days  ? 

Ans.  1  doll.  41 1  cts. 

3.  What  is  the  discount  of  600  dollars  for  90  days  ? 

Am.  9  dolls.  40  cts. 

4.  What  is  the  discount  of  1260  dollars  40  cents  for  60 
days  1  Ans.  13  dolls.  44  cts. 

5.  What  is  the  discount  of  2649  dolls.  75  cents  for  60 
days  ?  Ans.  28  dolls.  26  cts.  4  m. 

Form  of  a  note  offered  for  discount. 

Pittsburgh,  July  — ,  1832. 
Dollars 


Sixty  days  after  date,  I  promise  to  pay  A.  B.  or  order,  at 
the  bank  of ,  the  sum  of — dollars,  without  de- 
falcation ;  value  received.  J.  P. 


SECTION  8. 

EQUATION  OF  PAYMENTS. 

EQUATION  of  payments  is  the  finding  the  mean  time,  for 
the  payment  of  two  or  more  sums  of  money  payable  at  dif- 
ferv.~t  times. 

RULE. 

Multiply  each  sum  by  its  own  time.     Add  the  products 


94  FELLOWSHIP. 

into  one  sum  and  divide  this  amount  by  the  whole  debt ;  t/ie 
quotient  will  be  the  mean  time. 

EXAMPLE. 

1.  A  owes  B  600  dollars,  of  which  200  is  to  be  paid  at 
4  months,  200  at  8  months,  and  200  at  12  months;  but  they 
agree  to  make  but  one  payment ;  when  must  that  paymetit 
be  made? 

200  X    4=   800 

200  X    8=1600 

200X12  =  2400 


600  )  4800  (  8  months.    Ans. 

4800 


2.  A  merchant  has  owing  to  him  from  his  friend,  the  sum 
of  3000  dollars,  to  be  paid  as  follows,  viz.  500  dollars  at  2 
months,  1000  dollars  at  5  months,  and  t  le  rest  at  8  months  ; 
but  they  agree  to  make  one  payment  of  the  whole ;  whit 
will  be  the  mean  time  of  payment?  Ans.  6  months. 

3.  A  buys  of  B  50  acres  of  land,  for  which  he  agrees  to 
pay  1000  dollars  at  the  following  times,  viz.  200  dollars  at 
5  months,  300  dollars  at  8  months,  and  the  rest  at  10  months, 
but  an  equation  of  payments  is  afterwards  agreed  upon ; 
when  must  the  payment  be  made? 

Ans.  8  months  12  days. 

4.  C  owes  D  1400  dollars,  to  be  paid  in  3  months,  but  D 
being  in  want  of  money,  C  pays  him  1000  dollars  at  the  ex- 
piration of  2  months ;  how  much  longer  than  3  months  may 
he  in  justice  defer  the  payment  of  the  rest  ? 

Ans.  2%  months 


SECTION  9. 

FELLOWSHIP. 

'SHH' loaches  to  find  tho  profit  or  loss  arising  fo 
different  partners  in  trade,  in  proportion  to  the  cap:  J  or 
stock  each  has  advanced. 

Fellowship  is  either  nngle  or  compound. 


SINGLE  FELLOWSHIP.  95 

SINGLE  FELLOWSHIP, 

Is  when  the  stocks  employed  are  different,  but  the  time 
alike. 

RULE. 

Find  the  amount  of  the  whole  stock  employed ;  and  then 
(by  proporfton)  as  the  whole  stock  is  to  the  whole  gain  or 
loss,  so  is  each  partner's  stock  to  his  share  of  the  gain  or 
loss. 

EXAMPLE. 

1 .  Two  merchants  join  their  stock  in  trade ;  A  puts  in 
eOO  dollars,  and  B  puts  in  400  dollars,  and  they  gain  250 
dollars ;  what  part  belongs  to  each  ? 

A  600  A    1  nnn     0-n        600  to  A's  share  150  ^ 

B  400  ;  400  to  B's  100  !  ^^ 

1000  250 J 

2.  Three  merchants  enter  into  partnership  in  trade ;  A 
advanced  7500  dollars,  B  6000,  and  C  4500,  with  this  they 
gained  5400  dollars ;  what  was  each  partner's  share  ? 

C  A  2250  dolls, 
Ans.  I  B  1800 
I  C  1350 

3.  A  bankrupt  is  indebted  to  A  1291  dollars  23  cents,  to 
B  500  dollars  37  cents,  to  C  709  dollars  40  cents,  to  D  228 
dollars ;  and  his  estate  is  worth  but  2046  dollars  75  cents ; 
how  much  does  he  pay  per  cent,  and   how  much  is  each 
creditor  to  receive  ? 

D.     c. 

(A  receives  968  42$ 
B  375  27 | 

C  532  05 

D  171 

4.  Three  men,  A,  B,  and  C,  rent  a  farm  containing  585 
acres  2  roods  and  34  perches,  at  600  dollars  per  year,  oJ 
which  A   pays  180  dollars,  B  195,  and   C   225,  and  they 
agree  that  the  larm  shall    be  divided   in  proportion  to  the 
rents;  how  many  acres  must  each  man  have? 

A.     R.     P. 
;  share  is   175     2     34j 
Ans.  {  B's  190     1      17^ 

219     2     22] 


i  A's  share  is   175 
,  {  B's  190 

f  C's  219 


96  S1JNGLE  FELLOWSHIP. 

5.  Three  merchants  freighted  a  ship  with  2160  barrels  of 
flour,  of  which  960  barrels  belonged  to  A,  720  barrels  to  B, 
and  480  barrels  to  C ;  but  on  account  of  stormy  weather 
they  were  obliged  to  throw  900   barrels  overboard ;  how 
many  barrels  did  each  man  lose? 

i  A  lost  400  barrels. 
Ans.  IE          $00 
(C          200 

6.  Three  merchants  join  stock  in  trade ;  A  put  in  1260 
dollars,  B  840  dollars,  and  C  a  certain  sum ;  and  they  gained 
825  dollars,  of  which  C  took  for  his  part  275  dollars  ;  re- 
quired A  and  B's  part  of  the  gain,  and  how  much  stock  C 
put  in? 

( A         gained       330  dolls. 
<    Ans.  <B  220 

(  C's  stock  was    1050 

7.  Four  men  traded  with  a  stock  of  800  dollars,  and  they 
gained  in  two  years  time  twice  as  much,  and  40  dollars  over ; 
A's  stock  was  140  dollars,  B's  260,  C's  300  ;  required  D's 
stock,  and  what  each  gained  ? 

(D's  stock  was  100  dolls. 
A's  gain  was    287 
B's  533 

C's  615 

D's  205 

8.  Three  butchers  lease  a  pasture  field  for  96  dollars,  into 
which  they  put  300  beef  cattle  ;  of  these  80  belonged  to  A , 
100  to  B,  and  120  to  C ;  how  much  had  each  to  pay  ? 

Ans.  A  25  dolls.  60  cts.  B  32  dolls.  C  38  dolls.  40  cts. 

9.  A  father  left  an  estate  of  5000  dollars  to  his  three  sons, 
in  such  a  manner  that  for  every  2  dolls,  that  A  gets,  B  shall 
have  3,  and  C  5 ;  how  much  did  each  son  receive  ? 

Ans.  A  gets  1000  dolls.  B  1500,  C  2500. 

10.  A,  B  and  C  put  in  money  together,  A  put  in  20  dolls 
B  and  C  together  put  in  85  dollars ;  they  gained  63  dolls,  of 
which  B  got  21  dollars ;  what  did  A  and  C  gain,  and  how 
much  did  B  and  C  separately  put  in  ? 

{A  gained  12  dolls. 
C  30 

B  put  in    35 
C  50 


COMPOUND  FELLOWSHIP.  97 


COMPOUND  FELLOWSHIP. 

COMPOUND  FELLOWSHIP  is  when  both  the  stocks  'and 
times  are  different. 

RULE. 

Multiply  each  partner's  stock  by  the  time  it  is  employed, 
add  all  the  products  into  one  sum;  then  say,  as  the  sum  of 
the  products  is  to  the  whole  gain  or  loss,  so  is  each  part- 
ner's stock  multiplied  by  the  time,  to  his  share  of  the  gain 
or  loss. 

EXAMPLE. 

1.  Three  merchants  entered  into  trade;  A  put  in  2500 
dollars  for  4  months,  B  3000  dollars  fur  6  months,  and  C 
4000  dollars  for  8  months,  and  they  gained  1200  dollars; 
what  is  each  man's  share  of  the  gain  ? 

D.      M. 

A  2500x4=10000 
B  3000X6=18000 
04000X8=32000 


Sum  60000 

10000  :  200  A's  share  ) 

As  60000  :  1200  ::  <  18000  :  360  B's      V  Ans. 
32000  :  640  C's      > 


1200  proof. 

2.  Three  merchants  enter  into  partnership  for  16  months; 
A  put  into  stock  at  first  600  dollars,  and  at  the  end  of  8 
months,  200  dollars  more  ;  B  put  in  at  first  1200  dollars, 
but  at  the  end  of  10  months,  was  obliged  to  take  out  600 
dollars;  C  put  in  at  first  1000  dollars,  and  at  the  end  of  12 
months  put  in  800  more ;  with  this  stock  they  gained  2300 
dollars ;  what  was  each  man's  share  ? 

C  A's  share  is  560  dolls. 
Ans.  1  B's  780 

( C's  960 

3.  A  and  B  join  stock  in  trade ;  A  put  in  600  dollars  or. 
the  first  of  January ;  B  advanced  on  the  first  of  April  a  sum 


98  COMPOUND  FELLOWSHIP. 

which  entitled  him  to  an  equal  share  of  the  profit  at  the  end 
of  the  year ;  required  the  sum  B  put  in  ? 

Ans.  800  dollars. 

4.  D  put  in  stock  1800  dollars ;  E  at  the  end  of  4  months 
agrees  to  advance  such  a  sum  as  at  the  end  of  the  year  wili 
entitle  him  to  an  equal  share  of  .the  profits;  what  sum  must 
E  advance?  Ans.  2700  dollars. 

5.  Two  gentlemen,  A  and  B,  hired  a  carriage  in  Pitts- 
burgh to  go  to  Philadelphia,  and   return,  for  160  dollars, 
with  liberty  to  take  in  two  others  by  the  way.     When  at 
Philadelphia  they  took  in  C,  and  afterwards,  100  miles  from 
Pittsburgh,  they  took  in  D.  Now  allowing  it  to  be  300  miles 
from  Pittsburgh  to  Philadelphia,  and  also  that  each  man  pays 
in  proportion  to  the  distance  he  rode ;  it  is  required  to  tell 
how  much  each  must  pay  ? 

!A  pays  60  dolls. 
B  60 

C  30 

D  10 

160  proof. 

6.  Three  graziers  hired  a  piece  of  pasture  ground  for  145 
dolls.  20  cents ;  A  put  in  5  oxen   for  4£  months,  B  put  in  8 
oxen  for  5  months,  and  C  put  in  9  oxen  for  6£  months ;  how 
much  must  each  pay  ? 

Ans.  A  pays  27  dolls.  B  48  dolls,  and  C  70  dolls.  20  cts. 

7.  A,  B,  and  C  have  received  665  dollars  interest;  A  put 
in  4000  dolls,  for  12  months,  B  3000  for  15  months,  and  C 
5000  for  8  months ;  how  much  is  each  man's  part  of  the  in- 
terest ? 

(  A  240  dolls. 
Ans.  I  B  225 
f  C  200 

HI.  Three  merchants  lost  by  somo  dealings  263  dollars  90 
•-outs  ;  A's  stock  was  580  dolls,  for  6£  months,  B's  580  dolls, 
for  9J  months,  and  C's  870  dolls,  frr  8|  months ;  how  much 
is  each  man's  part  of  this  loss  ? 

A's  loss  59  dolls.  15  cts. 
Ans.      B's          86  45 

C's         118  30 


PROFIT  AND  LOSS.  99 

SECTION  10. 

PROFIT  AND  LOSS. 

BY  this  rule  we  discover  what  has  been  gained  or  lost 
on  the  purchase  and  sale  of  goods,  and  merchandise  of  every 
kind. 

RULE. 

Prepare  the  question  by  reduction  when  necessary,  and 
then  work  by  the  Rule  of  Three  or  Practice,  as  the  nature 
of  the  question  may  require. 

EXAMPLE. 

1.  Bought  360  barrels  of  flour  for  6  dollars  25  cents  per 
barrel,  and  sold  it  for  7  dollars  50  cents  per  barrel;  what 
is  the  profit  on  the  whole  ? 

D.    c. 

7     50 

6     25 


1     25  gain  per  barrel. 
B.       D.c.  B.  D. 

As    1      :     1   25     ::     360     :     450   Ajis. 

2.  Bought  a  piece  of  cloth  for  1  doll,  and  20  cents  pei 
yard,  and  sold  it  again  for  1  dollar  50  cents  a  yard ;  what 
is  the  gain  per  cent  1  Ans.  25  per  cent. 

3.  Bought  a  piece  of  linen  containing  42  yards  for  21 
dollars,  and  sold  it  at  66  cents  per  yard ;  what  is  the  gain 
or  loss  on  the  whole  piece  ? 

Ans.  6  dolls.  72  cents  gain. 

4.  A  merchant  bought  6  barrels  of  whiskey  containing 
32  gallons  each,  for  96  dollars ;  while  in  his  possession  he 
lost  12  gallons  by  leakage,  the  residue  he  sold  for  such  a 
sum  as  gained  him  12  dollars  on  the  whole;  how  much  per 
gallon  did  he  buy  and  sell  for  ? 

Ans.  Bought  for  50  cents,  and  sold  for  60  cents  per  gall. 

5.  Bought  120  doz.   of  knives  for  20  cents  each  knife, 
and  sold  them  again  for  17  cents  each,  what  was  the  loss  on 
the  whole?  Ans.  43  dolls.  20  cts. 


100  PROFIT  AND  LOSS 

6.  A  merchant  gave  149  dollars  for  100  yards  of  cloth  ; 
at  how  much  per  yard  must  he  sell  it  to  gain  51  dollars  on 
the  whole  1  Arts.  2  dollars. 

7.  Bought  a  chest  of  tea  at  1  dollar  and  25  cents  per 
pound,  but  finding  it  to  be  of  an  inferior  quality,  I  am  will- 
ing to  lose  18  per  cent,  by  it ;  how  must  I  sell  it  per  pound  ? 

Ans.  1  doll.  24  cents  per  Ib. 

8.  A  merchant  bought  20  dozen  of  wool  hats  at  90  cents 
per  hat ;  at  what  rate  must  he  sell  them  again  to  gain  20  per 
cent,  and  how  much  does  he  gain  on  the  whole  1 

Ans.  he  must  sell  at  1  dollar  8  cents  per  hat,  and  gains 
48  dollars  20  cents. 

9.  A  trader  bought  a  hogshead  of  rum  of  a  certain  proof, 
containing  115  gallons,  at  1  dollar  10  cents  per  gallon ;  how 
many  gallons  of  water  must  he  put  into  it  to  gain  5  dollars, 
by  selling  it  at  1  dollar  per  gallon  1 

Ans.  16i  gallons. 

10.  A  merchant  bought  4  hundred  weight  of  coffee  for 
134  dollars  40  cents,  and  was  afterwards  obliged  to  sell  it 
at  25  cents  per  pound ;  what  was  his  loss  on  the  whole,  and 
how  much  on  each  pound  ? 

Ans.  5  cents,  loss  on  each  pound,  and  22  dollars  40 
cents  on  the  whole. 

11.  If  by  selling  360  yards  of  broadcloth  for  1728  dollars, 
there  is  gained  20  per  cent,  profit,  what  did  it  cost  per  yard  ? 

Ans.  4  dollars. 

12.  A  merchant  laid  out  1000  dollars  on  cloth,  at  4  dol- 
lars per  yard,  and  sold  it  again  at  4  dollars  90  cents  per 
yard ;  what  was  his  \yhole  gain  ?  Ans.  225  dolls. 

13.  A  sells  a  quantity  of  wheat  at  1  dollar  per  bushel, 
and  gains  20  per  cent. ;  shortly  after  he  sold  of  the  same  to 
the  amount  of  37  dollars  50  cents,  and  gained  50  per  cent. ; 
how  many  bushels  were  there  in  the  last  parcel,  and  at  what 
rate  did  he  sell  it  per  bu  >hel  ? 

Ans.  30  bushels,  at  1  doll.  25  cents  per  bushel. 

14.  A  trader  is  about  purchasing  5000  galls,  of  whiskey, 
which  he  can  have  at  48  cents  per  gallon  in  ready  money, 
or  50  cents  with  two  months  credit ;  which  will  be  the  most 
profitable,  either  to  buy  it  on  credit,  or  by  borrowing  the 
money  at  8  per  cent,  per  annum,  to  pay  the  cash  price? 

Ans.  he  will  gain  68  dollars  by  paying  the  cash. 

15.  A  butcher   bought   12   head   of  beef  cattle  of  equal 


BARTER.  101 

weight,  for  240  dollars,  which  he  sells  again  for  4  cents  per 
pound ;  what  ought  each  one  to  weigh,  that  the  butcher  may 
have  the  hides  and  tallow  as  clear  gain  ? 

Ans.  4:cwt.  Iqr.  24Z&. 


SECTION  11. 

BARTER. 

BARTER  is  the  exchanging  of  one  commodity  for  another 

at  the  rates  agreed  upon  by  their  owners. 

$ 

RULE. 

Proceed  by  the  rules  of  reduction  and  proportion,  as  the 
nature  of  the  question  may  require. 

EXAMPLE. 

1.  How  many  yards  of  linen  at  50  cents  per  yard  must 
be  given  for  6^  yards  of  broadcloth,  at  4  dollars  50  cents 
per  yard  ? 

4,50  dollars 
6* 


2700 
112J 

28,12$ 

c.          yd.          D.   c.  yds. 

As    50     :     1      ::     28,12*     :     56%  Ans. 

2.  A  has  320  bushels  of  salt  at  1  dollar  20  cents  per 
bushel,  for  which  B  agrees  to  pay  him  160  dollars  in  cash 
and  the  rest  in  coffee  at  20  cents  per  pound ;  how  much  cof- 
fee must  A  receive?  Ans.  1120  Ib. 

3.  How  much  rye  at  70  cents  per  bushel  must  be  given 
for  28  bushels  of  wheat,  at  1  dollar  25  cents  per  bushel  ? 

Ans.  50  bushels. 

4.  A  barters  319  Ib.  of  coffee  at  23*  cents  per  pound, 
with  B  for  250  yards  of  muslin ;  what  does  the  muslin  cost 
A  per  yard  1  Ans.  30  cents  nearly. 

5.  C  has  flour  at  5  dollars  per  barrel,  which  he  barters 

12 


102  EXCHANGE. 

to  D  at  a  profit  of  20  per  cent,  for  tea  which  cost  1  dollar 
25  cents  per  pound ;  at  what  rate  must  D  sell  the  tea  to 
make  the  barter  equal?  Ans.  I  doll,  50  cts.  per  Ib. 

6.  A  has  cloth  which  cost  him  2  dollars  50  cents  per  yard, 
but  in  trade  he  must  have  2  dollars  80  cents ;  B  has  wheat 
at  1  dollar  20  cents  per  bushel ;  at  how  much  per  bushel 
should  he  sell  to  A,  to  make  the  barter  equal  ? 

Ans.  I  doll.  34f  cents. 

7.  P  has  240  bushels  of  rye  which  cost  him  90  cents  per 
oushel ;  this  he  barters  with  Q  at  95  cents  per  bushel  for 
wheat  which  stands  Q  99  cents  per  bushel ;  how  many  bushels 
of  wheat  is  he  to  receive  in  barter,  and  at  what  price,  that 
their  gains  may  be  equal  ? 

Ans.  218/y  bushels,  at  1  doll.  4|  cts.  per  bushel. 

8.  A  gives  B  in  barter  26lb.  4oz.  of  cinnamon,  at  1  dol- 
lar 28  cents  per  pound,  for  rice  at  6  cents  per  pound ;  how 
much  rice  must  A  receive  ?  Ans.  5  cwt. 

9.  C  and  D  barter ;  C  has  muslin  that  cost  him  22  cents 
per  yard,  and  he  puts  it  at  25  cents ;  D's  cost  him  28  cents 
per  yard  ;  at  what  price  must  he  put  it  to  gain  10  per  cent, 
more  than  C  ?  Ans.  34||  cents  per  yard. 

10.  A  buys  250  barrels  of  flour  from  B,  at  6  dollars  25 
cents  per  barrel,  in  payment  B  takes  4  cwt.  of  coffee  at  30 
cents  per  pound,  64  pounds  of  tea  at  1  dollar  75  cents  per 
Ib.  25  yards  of  broadcloth  at  6  dollars  per  yard,  206  dollars 
10  cents  in  cash,  and  the  balance  in  salt,  -at  8  dollars  per 
barrel ;  how  many  barrels  of  salt  must  B  receive  ? 

Ans.  120  barrels. 


SECTION  12. 

EXCHANGE. 

EXCHANGE  is  the  reducing  the  money,  coin,  &c.  of  one 
tate  or  country  to  its  equivalent  in  another. 

Par  is  equality  in  value ;  but  the  course  of  exchange  is 
often  above  or  below  pcu . 

Agiu  is  a  term  sometimes  used,  to  express  the  difference . 
between  bank  and  current  money. 

Case  1. 
To  change  the  currency  of  one  state  into  that  of  another. 


EXCHANGE 


RULE. 

Work  by  the  Rule  of  Three ;  or  by  the  theorems  in  the 
following  table : — 


*  The  New  England 
and  Maine. 
Note.  —  In  some  part* 
Louisiana,  Mississippi,  I 

GQ 

O 

f  ? 

°§.  §- 

~     p 
p 

r 

OKI 

5  2 
CL  ^ 

^i3 

o'  co 

!> 

"~i  P 

CD  £3 

R»?' 

^1 

II 

5*  P~* 
-    » 

0°  02 

ri 

3    p: 
Cta 

CD 

[TABLE,  exhibiting 
Theorems  fo 

p 

P  CD 

.0- 

sr 

CD 
02 

i  ,-*JL^ 

K 

>l  * 
Is   3 

i 

l^3 

CD    o- 

O  02 

P  C 
CD  cT 

»t 

|f|.^ 

s  ^** 

11    B 

«o  '->-' 

01  ^T 

co   r± 

^  2  s^  s 

^^   S 

P  CD        p 

*     P 

cr 

jj*    Q 

P^  O 

^.  re  a  ^ 

Ct5      ^5 

0    0>         "§ 

a 

•       <-^ 

p.  ^  s"  3" 

0 

|  1"      3' 



50  ^^f 

f! 

o4      S 

-<i— 

S    §' 

||  1 

?^^ 

§| 

<•£ 

o 

CD    t> 

^&^J 
a^&'S  s 
**  S       5 

j|  1 

•    P 

t—  *   J^ 
O5  p 

Sf 

*m 
Ril- 

ii 

*-"*        JO 

0- 

^    c^ 

o>  £i      sr 

C^ 

"S  •       £_ 

>  —  . 

53    S* 

s'S   ; 
'  ll   I 

BQ 

0^    rj-     (0 

QO  O^ 

D 

H-*  p- 

CJ  P- 

O 

g  >• 

New  York 
and 
Vorth  Carotin 

Z.    f? 

^  a- 

si 

s> 

^§  "* 

"i  § 

^H 

=!-   CC 

|; 

&1 

1  1 

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^2- 

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?  g" 

fifi 

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rT 

MH-P- 

s' 

S 

The  value  of  a  dollar  in  any  state  is  found,  either  opposite 
to  that  state,  or  under  it  in  the  table. 


104  EXCHAN^K. 


EXAMPLE. 

1.  What  is  the  value  of  480Z.  Pennsylvania  currency  in 
North  Carolina? 


s.    d. 

s. 

£. 

£. 

As 

7     6 

:     8     :: 

480 

:     512    Ans. 

£. 

Or, 

480 

Add  V 

t  =   32 

512  Ans. 

2.  What  is  the  value  of  256Z.  New  York  currency  in 

Pennsylvania?  Ans.  240Z. 

How  much  South  Carolina  currency  is  equal  to  1500Z. 

of  New  Jersey?  Ans.  933Z.  6s.  3d. 

4.  What  sum  New  York  currency  is  equal  to  180Z.  in 
Massachusetts?  Ans.  240Z. 

1.  How  much  Virginia  currency  will  purchase  a  bill  for 
280Z.  South  Carolina  ?  Ans.  360Z. 

6.  A  bill  of  exchangj  being  remitted  from  Rhode  Island 
__  Kjouth  Carolina  for  304Z.,  what  is  its  value  in  the  currency 
of  the  latter?  Ans.  236Z.  85. 


Case  2. 

To  change  the  currency  of  the  different  states  to  Federal 
money. 

RULE. 

Divide  the  given  sum,  reduced  to  shillings,  six-pences,  or 
pence  in  a  dollar,  as  it  passes  in  each  state. 

EXAMPLES. 

1.  Change  127Z.  12s.  New  England  money  to  dollars  and 
cents. 

127/.  12s.=2552  shillings. 
The  dollar,  New  England,  is  6s.  )  2552 

425,33£ 
Ans.  425  dolls.  33£  cts. 


FOREIGN   KXCHAIS'GK.  I0i» 

2.  Change  37/.  10s.  Pennsylvania  currency,  to  dollars. 
37  L  10s.  —  1500  six-pences. 

7s.  6d.  or  15  six-pences  make  a  dollar  Pennsylvania  cur- 
rency ;  hence  1500  ~  15=  100  dolls.  Ans. 


Or,  37Z.  10s.  =  9000  pence=  100,00  cents.  Ans. 

3.  Change   2251.  12s.  New  York  currency  to  Federal 
money. 

225L  12s.  —  4512  shillings  -h  8  the  dollar  New  York  cur- 
rency=564  dollars.  Ans. 

4.  A  bill  of  exchange  for  468Z.  9s.  6d.  Virginia  curren- 
cy, is  remitted  to  Philadelphia  ;  what  is  its  value  in  Federal 
money?  Ans.  1563  dollars  25  cts. 

5.  A  merchant  deposited  in  the   United  States    branch 
bank  at  Pittsburgh,  the  sum  of  750Z.  10s.  Pennsylvania  cur- 
rency, for  what  sum  may  he  draw  for  in  Federal  money  ? 

Ans.  2001  dollars  33^  cents. 

Note.  —  Federal  money  being  now  generally  introduced  into  mrrc;;r 
tile  business  throughout,  the  United  States,  has  nearly  superseded  ti": 
use  of  the  above  rules  of  exchange  between  the  different  States. 


Case  3, 
FOREIGN  EXCHANGE, 

Accounts  are  kept  in  Kngland,  Ireland,  and  the  West 
India  Islands,  in  pounds,  shillings,  pence,  and  farthings 
though  their  intrinsic  value  in  these  places  is  different. 

A  TABLE 

Of  different  Moneys^  as  they  are  denominated  and  valued 
in  different  countries. 

«REAT  BRITAIN,   IRELAND,   AND  THE  WEST  INDIES. 

4  farrhings  =1  penny 

12  pence  1  shillino 

20  shillings  I  pound 


106  FOREIGN  EXCHANGE. 


FRANCE. 

12 

Deniers 

=  1  Sol 

20 

Sols 

1  Livre 

3 

Livres 

-     1  Crown 

- 

SPAIN. 

4 

Marvadies  Vellon,  or 
Marvadies  of  Plate 

=  1  Quarta 

8* 
34 

Quartas,  or 
Marvadies  Vellon 

1  Rial  Vellon 

16 
34 

Quartas,  or 
Marvadies  of  Plate 

1  Rial  of  Plate 

8 

Rials  of  Plate 

1  Piaster,  Pezo,  or 

Dollar 

5 

Piasters    - 

1  Spanish  Pistole 

2 

Spanish  Pistoles    • 

-     1  Doubloon 

ITALY. 

12 

Deniers 

=  1  Sol 

20 

Sols 

1  Livre 

5 

Livres 

•     1  Piece  of  Eight  at 

Genoa 

6 

Livres 

1         Ditto       at  Leghorn 

6 

Solidi 

-     1  Gross 

24 

Grosses   - 

1  Ducat 

PORTUGAL. 

400 

Reas 

=  1  Crusadoe 

1000 

Reas 

-     1  Millrea 

HOLLAND. 

8 

Penning  - 

=  1  Groat 

2 

Groats 

-     1  Stiver  =2d. 

6 

Stivers 

I  Shilling 

20 

Stivers 

1  Florin,  or  Guilder 

3i 

Florins     - 

1  Rix  Dollar 

6 

Florins 

-     1  £.  Flemish 

5  Guilders  -  1  Ducat 

DENMARK. 

16    Shillings  -     =  1  Mark 

6  Marks ~  -     1  Rix  Dollar 
32     Rustics    -  1  Copper  Dollar 

6    Copper  Dollars     -         -     1  Rix  Dollar 

RUSSIA. 

18  Pennins  -     =  1  Gros 

30  Gros  -     1  Florin 

3  Florins     -  1  Rfx  Dollar 

2  Rix  Dollars  -     1  Gold  Puoai 


FOREIGN  EXCHANGE  ,10? 

• 

RULE, 

In  exchanging  of  foreign  moneys,  work  by  the  R  lie  of 
Three,  or  by  Practice ;  and  for  exchanging  foreign  moneys 
to  Federal,  work  by  the  table  in  page  40. 

EXAMPLES. 

1.  Philadelphia  is  indebted  to  London  1749/.  currency, 
what  sum  sterling  must  be  remitted,  when  the  exchange  is 
65  per  cent.  ? 

£.          £.  £.  £. 

As    165     :     100    ::    1749     :     1060  sterling.    Ans. 

2.  London  is  indebted  to  Philadelphia   1060Z.  sterling; 
what  sum  Pennsylvania  currency  must  be  remitted,  the  ex- 
change being  65  per  cent,  as  above  ? 

£.  £.  £.  £. 

As     100     :     165     ::     1060     :     1749     Ans. 

Or,     50     J     1060 

10     i       530 

5     |       106 

53 

1749Z.  Ans. 

3.  Baltimore,  Oct.  1,  1817. 
Exchange  for  1260Z.  10s.  sterling. 

Thirty  days  after  sight  of  this  my  first  of  exchange,  sec- 
ond  and  third  of  like  tenor  and  date  not  being  paid,  pay  to 
A.  B.  or  order,  twelve  hundred  and  sixty  pounds  ten  shil- 
lings sterling,  value  received,  and  place  the  same  to  account, 
as  per  advice  from 

P  —  S— n. 

W.  L.  merchant,  London. 

What  is  the  value  of  this  bill  in  Federal  money  ? 

1260Z.  1  Os.  =  1260,5.x  by  444  cents=5596  dolls.  Oa 
cents.     Ans. 


108  FOREIGN  EXCHANGE. 

• 

4.  London,  January  1,  1818. 

Exchange  for  5596  dolls.  62  cts.  Federal  money. 
Thirty  days  after  sight  of  this  my  second  of  exchange, 
first  and  third  of  the  same  tenor  and  date  not  paid,  pay  to 
J.  B.  or  order,  five  thousand  five  hundred  and  ninety -six 
dollars  sixty-two  cents,  value  received,  and  place  the  same 
to  account,  as  per  advice  from 

S.'S. 

Mr.  T.  L.  merchant,  Baltimore. 

How  much  sterling  is  the  above  bill,  4,44  cents  to  the 
pound? 

444)5596,62(1260 
444 


1156 

888 

2686 
2664 

222 

20 


4440(10     Ans.  1260Z.  10*. 
4440 


5.  A  merchant  of  Philadelphia   receives  from  his  cor- 
respondent in  Dublin,  a  bill  of  exchange  for  540Z.  15s.  Irish 
currency ;  what  is  its  value  in  Federal  money  ? 

Ans.  2217  dolls.  7£  cts. 

6.  A  merchant  in  Philadelphia  draws  on  his  correspond- 
ent in  Dublin  for  the  balance  of  an  account  amounting  to 
2217  dolls.  7£  cents ;  what  sum  Irish  currency  mast  be  re- 
mitted to  satisfy  the  draft?  Ans.  540/.  15s. 

Note. — In  ti»Prie  last  examples  tha  course  of  exchange  is  considered 
;IH  1  cin{r  at  jKir:  when  the  exch.-mge  is  nbove  or  below  par,  the  per 
cent,  must  be  added  or  subtracted,  as  the  case  requires. 

7.  In  a  settlement  between  A  of  London  and  B  of  Phila- 
delphia, B  is  indebted  to  A  in  the  sum  of  3207.  sterling,  what 
sum  must  be  remitted  by  B  to  A  to  settle  the  balance,  the 
uxv  hange   being   12J   per  cent,   from   the  United  States  to 
(iivat  Britain?  Ans.   1598  dolls.  40  cts. 


ALLIGATION.  109 

8.  C  of  New  Yo]  ;v  remits  3259  dollars  to  his  correspond- 
ent in  Dublin,  to  be  placed  to  his  account ;  for  what  sum 
Irish  currency,  must  he  receive  credit,  the  course  of  ex- 
change being  8  per  cent,  in  favor  of  Ireland  ? 

Ans.  7361.  nearly. 

Note.  The  par  ?f  exchange  between  the  United  States  of  America 
and  most  other  trading-  countries,  may  be  found  by  the  table  in  page  40. 


SECTION  13. 

ALLIGATION. 

ALLIGATION  is  a  rule  for  finding  the  prices,  and  quantity 
of  simples  in  any  mixture  compounded  of  those  things.* 

Case  1. 

To  find  the  mean  price  of  any  part  of  the  composition, 
when  the  several  quantities  and  prices  are  given. 

RULE. 

As  the  sum  of  the  whole  quantity,  is  to  its  total  value,  so 
is  any  part  of  the  composition,  to  its  value. 

EXAMPLE. 

1.  A  merchant  mixed  2  gallons  of  wine  at  2  dollars  per 
gallon,  2  at  2  dollars  50  cents,  and  2  at  3  dollars ;  what  is 
one  gallon  of  this  mixture  worth  ? 
gal. 

2  at  2,00^400 
2  at  2, 50 = 500 
2  at  3,00  =  600 


6  1500 

G.         D.  c.  G.        D.c. 

As  6     :     15,00     ::     1      :     2,50  Ans. 

2.  A  grocer  mixed  20  Ib.  of  sugar  at  10  cents  per  Ib.  30 
Ib.  at  15  cents,  and  40  Ib.  at  25  cents;  what  is  one  pound 
of  this  mixture  worth?  Ans.  18-1-  cts. 

3.  A  trader   mixes  10  bushels  of  salt  at  150  cents,  20  at 

K 


110  ALLIGATION. 

160  cts.  and  30  at  170  cts.  per  bushel;  at  what  rate  can  he 
afford  to  sell  one  bushel  of  this  mixture?       Ans.  1G3-J  cts. 

4.  If  4  ounces  of  silver  at  75  cents  per  ounce,  be  melted 
with  8  ounces  at  60  cents  per  ounce,  what  is  the  value  of 
one  ounce  of  this  mixture  ?  Ans.  65  cents. 

Case  2. 

To  find  what  quantity  of  several  simples  must  be  taken 
at  their  respective  rates,  to  make  a  mixture  worth  a  given 
price. 

RULE. 

Place  the  rates  of  the  simples  under  each  other,  and  link 
each  rate  which  is  less  than  the  mean  rate,  with  one  or  more 
that  is  greater.  The  difference  between  each  rate  and  the 
mean  price  set  opposite  to  the  respective  rates  with  which  it 
is  linked,  will  be  the  several  quantities  required. 

Note.  1.  If  all  the  given  prices  be  greater  or  less  than  the  mean 
rate,  they  must  be  linked  to  a  cipher. 

2.  Different  modes  of  linking  will  produce  different  answers. 

EXAMPLES. 

1.  How  many  pounds  of  tea  at  150,  160,  and  200  cents 
per  pound,  must  be  mixed  together,  that  1  pound  may  be 
sold  for  180  cents? 

t  150x  20  at  150  cents  } 

Mean  rate  180  2  160\)  20  at  160  V  Ans. 

(  200^  30  +  20=50  at  200  ) 

2.  How  many  gallons  of  wine  at  3,  5,  and  6  dollars  per 
gallon,  must  be  mixed   together,   that  one  gallon  may  be 
worth  4  dollars? 

Ans.  3  gallons  at  3  dolls.  1  gallon  at  5  dolls,  and  1 
gallon  at  6  dollars. 

3.  How  many  bushels  of  rye  at  40  cents  per  bushel,  and 
corn  at  30  cents,  must  be  mixed  with  oats,  at  20  cents,  to 
make  a  mixture  worth  25  cents  per  bushel  ? 

t  20^rx  15  f  5  C    6  bushels  of  rye 

1.  Ans.  25  ?  30-J  )  5         2.  Ans.  <    6        do.        corn 

(  40-^  5  (24        do.         oats 

4.  A  grocer  has  four  several  sorts  of  tea,  viz.  one  kind 
at  120  cents,  another  at  110  cents,  another  at  90  cents 


ALLIGATION.  Ill 

and  another  at  80  cents  per  pound,  how  much  of  each  sort 
must  be  taken  to  fnake  a  mixture  worth  1  dollar  per  pound  ? 
'  2  at  120  cents.  (  3  at  120  cents. 

110  2  ^ns    >2      110 

(3       80 

f  1  at  120  cents. 

(  1        80 
f  2  at  120  cents. 
6.  Ans.  )  j*       *g0 
(3         80 

Note.  From  this  last  example  it  is  manifest  that  a  great  many  dif- 
ferent answers  may  result  to  the  same  question,  according  to  the  va- 
rious modes  of  linking  the  numbers  together. 

Case  3. 

When  the  rate  of  all  the  simples,  the  quantity  of  one  of 
them,  and  the  compound  rate  of  the  whole  mixture  are  given, 
to  find  the  several  quantities  of  the  rest. 

RULE. 

Arrange  the  mean  rate,  and  the  several  prices,  linked  to- 
gether as  in  case  2,  and  take  their  difference. 

Then,  as  the  difference  of  the  same  name  with  the  quan- 
tity given, 

Is  to  the  rest  of  the  differences  respectively  : 

So  is  the  quantity  given, 

To  the  several  quantities  required. 

EXAMPLE. 

1.  A  grocer  would  mix  40  pounds  of  sugar  at  22  cents 
per  pound,  with  some  at  20,  14,  and  12  cents  per  pound  ; 
how  much  of  each  sort  must  he  take  to  mix  with  the  40 
pounds,  that  he  may  sell  the  mixture  at  1 8  cents  per  pound  ? 
f!2— N  — 41b. 

is)  14    V    ~2 

^  20_  /  /      4 

.  22—          — 6  against  the  price  of  the  given  quantity. 
As  6      :     40     ::     4      :     26,66  Ib.  at  12  cents.) 

6      :     40      ::     2      :      13,33  do.       14  }   Ans. 

and  26,66  do.      20 


112  ALLIGATION. 

2.  How  much  wheat  at  48  cents,   rye  at  06  cents,  and 
barley  at  30  cents  per  bushel,  must  be  mixed  with  24  bush- 
els of  oats  at  18  cents  per  bushel,  that  the  whole  may  rate 
at  22  cents  per  bushel  ?  Ans.  2  bushels  of  each. 

3.  How  much  gold  at  16,  20,  and  24  carats  fine,  and  how 
much  alloy  must  be  mixed  with  10  ounces  of  18  carats  fine, 
that  the  composition  may  be  22  carats  fine  ? 

Ans.  10  oz.  of  16  carats  fine,  10  of  20,  170  of  24, 
and  10  of  alloy. 

Case  4. 

When  the  price  of  all  the  simples,  the  quantity  to  be  mix- 
ed, and  the  mean  price  are  given,  to  find  the  quantity  of  each 
simple. 

RULE. 

Find  their  differences  by  linking  as  before : 
Then,  as  the  sum  of  the  differences, 
Is  to  the  quantity  to  be  compounded  ; 
So  is  the  difference  opposite  to  each  price, 
To  the  quantity  required. 

EXAMPLE. 

1.  How  much  sugar  at  10,  12,  and  15  cents  per  pound, 
will  be  required  to  make  a  mixture  of  40  pounds,  worth  1  3 
cents  per  pound  ? 


8  sum  of  the  different  simples. 
As     8     :     40     ::     2     :     10  lb.  at  10  els.  ) 

8     :     40     ::     4     :     20  do.       15         \  Ans. 
and     10  do.      12         > 

2.  How  much  golJ  of  15,  of  17,  of  18,  and  of  22  carats 
fine,  must  be  mixed  together  to  form  a  mixture  of  40  ounces 
of  20  carats  fine? 

Ans.  5  oz.  of  15,  of  17,  and  of  1  8,  and  25  oz.  of  22. 

3.  How  many  gallons  of  water  must  be  mixed  wkh  wine 
at  (>  dollars  per  gallon,  to  fill  a  vessel  of  70  #,i lions,  so  that 
it  may  he  sold  without  loss  at  5  dollars  per  gallon? 

4/7A-.   1  1  ••  07)  lions  of  water. 


VULGAR  FRACTIONS.  1 1 3 

PART  VI. 
VULGAR  FRACTIONS. 

A  VULGAR  FRACTION  is  any  supposed  part  or  parts  of  an 
unit,  and  is  represented  by  two  numbers  placed  one  above 
the  other,  with  a  separating  line  between  them  ;  thus,  |  one- 
fifth,  £  four-ninths. 

The  number  above  the  line  is  called  the  numerator,  and 
that  below  the  line  the  denominator.  Thus, 

4  numerator  6 
—  —  &c. 

9denomina.  10 

The  denominator  shows  how  many  parts  the  unikpr  inte- 
ger is  divided  into,  and  the  numerator  shows  how  many  of 
those  parts  are  contained  in  the  fraction. 

Vulgar  fractions  are  either  proper,  improper,  compound, 
or  mixed. 

A  proper  fraction  is  when  the  numerator  is  less  than  the 
denominator,  as  -J,  f,  j,  |f ,  &c. 

An  improper  fraction,  is  when  the  numerator  is  eitner 
equal  to,  or  greater  than  the  denominator,  as  f ,  f,  2T°,  &c. 

A  compound  fraction  is  a  fraction  of  a  fraction,  as  f  of  $, 
f  of/0  of  jf,&c. 

A  mixed  fraction  is  a  whole  number  and  fraction  united, 
as  8|,  4J,  120f ,  &c. 


SECTION  1. 

Reduction  of  Vulgar  Fractions. 

Case  1. 
To  Reduce  a  Vulgar  Fraction  to  its  lowest  terms. 

RULE. 

Divide  the  greater  term  by  the  less,  and  that  divisor  by 
the  remainder,  till  nothing  be  left,  the  last  divisor  is  the 
K2 


114  REDUCTION  OF  VULGAR  FRACTIONS. 

common  measure,  by  which  divide  both  parts  of  the  frac- 
tion :  the  quotient  will  be  the  answer.    Or, 

Take  aliquot  parts  of  both  terms  continually,  till  the  frac- 
tion is  in  its  lowest  terms. 

Note.  1.  1.'  the  common  measure  when  found  is  1,  the  fraction  ia 
already  in  its  lowest  terms. 

2.    Ciphers   to  the  right  of  both  the  terms    may  be  cut  off  thus, 

w=f 

EXAMPLES. 

1.  Reduce  J|  to  its  lowest  terms. 
36  )  48 ( i 
36 

Common  measure      1 2  )  36  (  3 
36 

12)36(3 

36  3  2  div.  6  div. 

—  Ans.  36     18     3 

12)48(4  Or,— •=— =-    Ans. 

48  48     24     4 

2..  Reduce  ^f  £  to  its  lowest  terms.  Ans.  f 

3.  Reduce  T7^  to  its  lowest  terms.  | 

4.  Reduce  fVVVV  to  its  lowest  terms.  f 

5.  Reduce  TyT  to  its  lowest  terms.  | 

6.  Reduce  gW/T  to  its  lowest  terms.  J 

Case  2. 
To  reduce  a  mixed  number  to  an  improper  fraction. 

RULE. 

Multiply  the  whole  number,  by  the  denominator  of  the 
fraction,  and  add  the  numerator  to  the  product,  for  a  new 
numerator,  under  which  place  the  given  denominator. 

EXAMPLE. 

1.  Reduce  8$  to  an  improper  fraction. 
8 
4 

3-35 

— -  Ans. 
4 


REDUCTION  OF  VULGAR  FRACTIONS.       115 

2.  Reduce  12Tf  to  an  improper  fraction.  Ans.  2j~? 

3.  Reduce  183  -/T  to  an  improper  fraction.  *ff8 

4.  Reduce  514T\  to  an  improper  fraction.  8T|9 

5.  Reduce  68425?  to  an  improper  fraction.  a  7 3T7 ° 3 

Case  3. 

To  reduce  an  improper  fraction  to  a  whole  or  mixed 
number. 

RULE. 

Divide  the  numerator  by  the  denominator  ;  the  quotient 
will  be  the  answer  required. 
Note.     This  case  and  case  2,  prove  each  other. 

-     EXAMPLE. 

1 .  Reduce  \5  to  its  proper  terms. 

4)35(8|    Ans. 
32 

3 

2.  Reduce  3fT8  to  its  proper  terms.  Ans.   183^ 

3.  Reduce  2  \6  5  to  its  proper  terms.  352i 

4.  Reduce  3T6y   to  its  proper  terms.  56—y 

5.  Reduce  8y|9  to  its  proper  terms.  •                     514T5¥ 

Case  4. 

To  reduce  several  fractions  to  others  that  shall  have  one 
common  denominator,  and  still  retain  the  same  value. 

RULE. 

Reduce  the  given  fractions  to  their  lowest  terms,  then 
multiply  each  numerator  into  all  the  denominators,  but  its 
own.  for  a  new  numerator  ;  and  all  the  denominators  into 
each  other  for  a  common  denominator. 

EXAMPLE. 

1.    Reduce  t,  |,  and  |,  to  a  common  denominator. 
1X3X4—12V 
2X2X4— 16>  numerators. 
3X2X3—18) 
2X3X4—24     common  denominator. 


116 


REDUCTION  OF  VULGAR  FRACTIONS. 


2.  Reduce  f ,  |,  and  §-,  to  a  common  denominator. 

Ana.  ?&,  V&,  || 
8.  Reduce  i,  -|,  T4j,  and  f ,  to  a  common  denominator. 


Case  5. 

To  reduce  several  fractions  to  others,  retaining  the  same 
value,  and  that  shall  have  the  least  common  denominator. 

RULE. 

Divide  the  given  denominators  by  any  number  that  will 
divide  two  or  more  of  them  without  a  remainder ;  set  the 
quotients  and  undivided  numbers  underneath  ;  divide  these 
numbers  in  the  same  manner,  and  continue  the  operation, 
till  no  two  numbers  are  left  capable  of  being  lessened  ;  the 
product  of  these  remaining  numbers,  together  with  the  di- 
visor or  jdivisors,  will  give  the  least  common  denominator. 

Divide  the  common  denominator,  so  found,  by  each  par- 
ticular denominator,  and  multiply  the  quotient  by  its  own 
numerator  for  a  new  numerator,  under  which  place  the  com- 
mon denominator. 

EXAMPLE. 

1.  Reduce  |,  J,  j-,  and  f ,  to  the  least  common  denom- 
inator. 

3)2     368 


2)2     128 

111     4  X~2x  3=24  common  denominator. 
'2)24 


Divisors 


12X1  =  12 
3      8x2=16 
6      4X5=20 
^8      3X7=21 

Then,  if,  ||,  ff,  f i  Ans. 

2.  Reduce  f ,  f ,  T\,  and  vV,  to  the  least  common  donom. 
mator.  Ans.  A°n-,  rV5o>  TV« 


REDUCTION  OF  VULGAR  FRACTIONS.    _  117 

3.  Reduce  |,  f ,  T45-,  and  |,  to  the  least  common  denomi- 
nator. Ans.  J-f ,  H,  H>  If 

Case  6. 
To  reduce  a  compound  traction  to  a  single  one. 

RULE. 

Multiply  all  the  numerators  together  for  a  new  numerator, 
and  all  the  denominators  for  a  new  denominator. 

Note.  Such  figures  as  are  alike  in  the  numerators  and  denominators 
may  be  cancelled. 

EXAMPLE. 

1.  Reduce  f  of  J  of  £  to  a  single  fraction. 

2  X  3  X  4—  24         2  Or  cancelled. 

—  =   -  Ans.  2     3     4         2 

3X4X5=60          5  _     _     _  =   _  Ans. 

3     4     5         5 

2.  Reduce  J  of  £  of  T\  to  a  single  fraction. 

Ans.  fi 

3.  Reduce  £  of  f  of  J  to  a  single  fraction. 

Ans.  ^ 

4.  Reduce  f  of  ^  of  f  J  to  a  single  fraction. 

Ans.  TVo 
Case  7. 

To  reduce  a  fraction  of  one  denomination  to  the  fraction 
.of  another,  but  greater,  retaining  the  same  value. 

RULE. 

Make  it  a  compound  fraction,  by  comparing  it  with  all 
the  denominations  between  it  and  that  to  which  it  is  to  be 
reduced  ;  reduce  this  fraction  to  a  single  one. 

EXAMPLE. 

1 .   Reduce  f  of  a  penny  to  the  fraction  of  a  pound. 
5    X     1     X    1  5 


6   X    12  X  20          1440 

2.  Reduce  £  of  a  pennyweight  to  the  fraction  of  a  pound 
troy  Ans.  ¥JT 

%  Reduce  T9:i-  of  a  pint  of  wine  to  the  fraction  of  a  hogs- 
head. Ans.  ^fa 


118  REDUCTION  OF  VULGAR  FRACTIONS. 

4.     Reduce  TT  of  a  minute  to  the  fraction  of  a  day. 


TJ84 

Case  8. 

To  reduce  tne  fraction  of  one  denomination  to  the  frac- 
tion of  another,  but  less,  retaining  the  same  value. 

RULE. 

Multiply  the  given  numerator,  by  the  parts  of  the  denom- 
inator, between  it  and  that  to  which  it  is  to  be  reduced,  for  a 
new  numerator,  and  place  it  over  the  given  denominate,,-, 
which  reduce  to  its  lowest  terms. 


1.  Reduce  TT5T-g-  of  a  pound  to  the  fraction  of  a  penny. 

5     X20X12         1200         5 

1440X1  X  1  H40         6 

2.  Reduce  -$~  of  a  pound  troy  to  the  fraction  of  a  pen- 
nyweight. Ans.  j 

3.  Reduce  7~  of  a  hogshead  to  the  fraction  of  a  pint. 

Ans.    ^ 

4.  Reduce  yj-g-4  °f  a  day  to  tne  fraction  of  a  minute. 

Ans.  VT 
Case  9. 

To- find  the  value  of  a  fraction  in  the  known  paits  of  an 
integer. 

RULE. 

Multiply  the  numerator  by  the  known  .parts  of  the  inte- 
ger, and  diviilo  by  the  denominator. 

EXAMPLE. 

1.    What  is  the  value  of  f  of  a  pound  sterling? 
20  shillings  =  1  pound. 
2 

3  )  40 

13  4  Ans.    13*.  4ct 

t.    Red  are  J  of  a  pound  troy  tc  iu  proper  quantity. 

Ans.  7oz.  4dwt. 


REDUCTION  OF  VULGAR  FRACTIONS.  11D 

3.  Reduce  f  of  a  mile  to  its  proper  quantity. 

Ans.  6fur*.  Wp. 

4.  Reduce  T3F  of  a  day  to  its  proper  time. 

Ans.  7h.  I2min. 

5.  What  is  the  value  of  f  of  a  dollar.  Ans.  80  cts. 

Case  10. 

To  reduce  any  given  quantity,  to  the  fraction  of  a  greater 
Munomination  of  the  same  kind. 

RULE. 

Reduce  the  given  quantity  to  the  lowest  denomination 
mentioned  for  a  new  numerator,  under  which  set  the  integral 
part  (reduced  to  the  same  name)  for  a  denominator. 

EXAMPLES. 

1.  Reduce  6s.  Sd.  to  the  fraction  of  a  pound. 

s»      d.  s. 

68  20 

12  12 

80     1  240 

=-  Ans. 

240     3 

2.  Reduce  25  cents  to  the  fraction  of  a  dollar. 

25      1 

100     4 

3.  Reduce  31  gallons  2  quarts  to  the  fraction  of  a  hogs- 
head. Ans.  i. 

4.  Reduce  6  hundred  weight  2  quarters  18|  pounds  to  the 
fraction  of  a  ton.  Ans.  J. 

Case  11. 

To  reduce  a  vulgar  fraction  to  a  decimal  of  the  same 
value. 

RULE. 

Add  ciphers  to  the  right-hand  of  the  numerator,  and  cfivide 
by  the  denominator. 


120          •        ADDITION  OF  VULGAR  FRACTIONS 


EXAMPLE. 

'  1.   Reduce  f  £ o  a  decimal  fraction  of  the  same  value. 
4  )  300 

,75  Ans. 
2.  Reduce  ^J  to  a  decimal  fraction.  Ans.  ,85 


SECTION  2. 
ADDITION  OF  VULGAR  FRACTIONS. 

Case  1. 

To  add  fractions  that  have  the  same  common  denomi- 
nator. 

RULE. 

Add  all  the  numerators  together,  and  divide  the  amount 
by  the  common  denominator. 

EXAMPLE. 

1.  Add  T^,  r52,  T72>  T92  and  |J  together. 

numerators. 
1 
5 
7 
9 
11 

common  denominator  12  )  33  (  2|  An*. 
24 

9      3 

12     4 

2.  Add  /y,  T8T,  i j,  £{,  and  if  together.  Anj.  2$ 
8.  Add  Jf,  H,  AJ,and  JJ  together.  3*. 


ADDITION  OF  VULGAR  FRACTIONS.  121 

Case  2. 

To  add  fractions  having  different  denominators. 
RULE. 

Reduce  the  given  fractions  to  a  common  denominator,  by 
case  5,  and  proceed  as  in  the  foregoing  case. 

EXAMPLE. 

1.  Add  $,  J,  £  and  T9j  together. 

12 

1359  12  18  15  18  18 
----  =  --  ---  15 
I  4  8  12  24  24  24  *4  18 

24)63(2} 

48 

15 

2.  Add  i,  i,  £  and  £  together.  Ans*  1  & 

3.  Add  f  ,  f  ,3,  f  and  r85  together.  3^ 

Case  3. 

To  add  mixed  numbers. 

RULE. 

Add  the  fractions  as  in  the  foregoing  cases,  and  the  inte- 
gers as  in  addition  of  whole  numbers. 

EXAMPLES. 
I.  Add  13^,  97<y  and  3/j  together. 


2.  Add  5|,  6  j  and  4£  together. 

5|-rr5if  common  denominator 
6J  =  6U  15 


122  SUBTRACTION  OF  VULGAR  FRACTIONS. 

3.  Add  1J,  j  of  J,  and  9^  together.  Ans. 

4,  Add  I  fa  6£,  §  of  i,  and  7i  together. 

Case  4. 
To  add  fractions  of  several  denominations. 

RULE. 

Reduce  them  to  their  proper  quantities  by  case  10  in  re- 
duction,, and  add  them  as  before. 

EXAMPLE. 

1 .  Add  J  of  a  £.  and  y3^  of  a  shilling. 

s.    d. 

f  of  a  £.  =  15  6|=||  common  denom. 
ft  of  a    ,.-0  3j  =  A 


9 


Shillings    15  10/3-  Ans. 

2.  Add  i  of  a  yard  to  f  of  a  foot.    Ans.  2  feet  2  inches. 

3.  Add  i  of  a  day  to  £  of  an  hour. 

Ans.  8  hours  30  minutes. 

4.  Add  J  of  a  week,  J  of  a  day.  and  \  of  an  hour  to- 
gether.        .  Ans.  2  days  14  hours  30  minutes. 

5.  Add  J  of  a  miie,  f  of  a  yard,  and  J  of  a  foot  together 

Ans.  1540  yards  2  feet  9  inches. 


SECTION  3. 

Subtraction  of  Vulgar  Fractions. 

RULE. 

PREPARE  the  fractions  as  in  addition,  and  sub:ract  the 
rower  numerator  from  the  upper,  and  place  the  rfrilerence 
over  the  common  denominator. 

Note.  1.  When  the  lower  numerator  is  greater  than  the  upper,  sub- 
tract it  from  the  common  denominator,  adding  the  .nipper  numerator  to 
*he  difference,  and  carry  1  tcf  the  units  place  of  t^e  integer. 

2.  When  the  fractions  are  of  different  integers,  find  their  values, 
separately t  and  subtract  as  in  compound  subtraction  of  whole  tium 


MULTIPLICATION  OF  VULGAR  FRACTIONS. 


123 


EXAMPLES. 


From  | 
Take  f 


From  f 
Take 


From  1= 
Take  i= 


Rem.  f =i  A/is.       Rem. 


From  ji 
Take   J 

Rem.  i 


From 
Take 

Rem. 


From  f 
Take  j 

Rem.  TjV 

From  13 
Take     6 


Ans.       Rem. 
From  |?f 


Rem. : 

From 
Take 

Rem. 


•fc  Ans. 

-  From  }| 
Take  |i 

Rem.  TV 

From  19T\ 
Take    0T7 


Rem. 


From  Jof  a  £.=15     6| 
Take  ft  of  a  s.  =  0     3J 


Rem. 


15 


From  7      weeks 
Take  9TV  days 


Rem. 


.  7h.  12m. 


SECTION  4. 

Multiplication  of  \  ulgar  Fractions. 

RULE. 

REDUCE  the  compound  fractions  to  simple  ones,  and 
mixed  numbers  to  improper  fractions,  then  multiply  the  nu- 
merators together  for  a  new  numerator,  and  the  denomina- 
tors for  a  new  denominator. 


1.  Multiply  J  by 


EXAMPLES. 
2X1=2         1 

-    -    —  =-  An*. 
3X4=12     0 


124  DIVISION7  OF  VTLHAR  FRACTIONS. 

2.  Multiply  4i  by  | 

2 

9X1=  9 

-     -     —  An*. 

2x8=16 

3.  Multiply  |  by  f Ans.  ^ 

4.  „        '    J  of  |  by  £  .         .          .          -          & 

5.  H  by  i  -         •         .         -  1  j 

T.  48}  by  13|         -         .         .         .         672y\ 


SECTION  5. 

Division  of  Vulgar  Fractions. 

RULE. 

PREPARE  the  fractions,  if  necessary,  by  reduction ;  inrert 
the  divisor,  and  proceed  as  in  multiplication. 

EXAMPLES. 

1.  Divide  f  by  j| 

3x3      9 

-    -=—  An*. 

8x2     16 

2.  Divide  4i  by  1| 

*J  H 

2  3 

9X3     27 
0  5         Then  -     -=—=2,7, 

2X5     10 
2  a 


RULE  OF  THREE  IN  VULGAR  FRACTIONS.  125 

3.  Divide  f  by  f  Ans.  f  } 

4.  Divide  if  by  f  Iff 

5.  -Divide  li  by  4T\  ft 

6.  Divide  J  by  4  /, 

7.  Divide  9|  by  i  of  7  2if 

8.  Divide  5205}  by  f  of  91  71* 


SECTION  6. 

The  Rule  of  Three  in  Vulgar  Fractions. 

THE  operation  of  tbe  Rule  of  Three  in  Vulgar  Fractions, 
whether  direct,  inverse,  or  compound,  is  performed  <in  the 
same  manner  and  agreeably  to  the  principles  laid  down  in 
whole  numbers  under  these  rules. 

When  the  question  is  in  direct  proportion,  prepare  the 
terms  by  reduction,  and  invert  the  first  term ;  then  proceed 
as  in  multiplication  of  fractions. 

EXAMPLE. 

1.  If  i  of  a  yard  of  cloth  cost  f  of  a  dollar,  what  will  I 
of  a  yard  come  to  ? 

yd.        D.         yd. 
As    i     :     1     ::     | 
4X2X7  =  56  D.c. 

Inverted  -    -    -    — =2J  dollars.      Ans.  2  33 J 
1X3X8  =  24 

2.  If  f  of  a  ton  of  iron  cost  164J  dollars,  what  will  ^  of 
a  ton  come  to?  Ans.  211  dolls.  28|  cts. 

3.  A  person  having  f  of  a  coal  mine,  sells  |  of  his  share 
for  171  dollars,  what  is  the  value  of  the  whole  mine  at  the 
same  rate?  Ans.  380  dollars. 

4.  At  f  of  a  dollar  per  yard,  what  will  42  yards  come  to  ? 

Ans.  35  dollars. 

5.  A  gentleman  owning  |  of  a  vessel,  sells  f  of  his  share 
for  312  dollars,  what  is  the  whole  vessel  worth  ? 

Ans.  11 70  dollars. 
L2 


126  INVERSE  PROPORTION. 

6.  If  li  bushel  of  apples  cost  79 J  cents,  what  will   3f 
bushels  cost  at  the  same  rate  ?  -  Ans.  202T\  cents. 

7.  If  J  of  a  ship  be  worth  175  dollars  35  cents,  what 
part  of  her  may  be  purchased  for  601  dollars  20  cents? 

Ans.  3j 


SECTION  7. 
INVERSE  PROPORTION. 

RULE. 

PREPARE  the  question  as  in  direct  proportion,  invert  the 
third  term,  and  proceed  as  in  multiplication  of  fractions. 

EXAMPLES. 

1.  How  much  shalloon  f  yard  wide,  will  lino  4J  yards  of 
cloth  l£  yard  wide? 

H  =  %  Then,  as  I  :  f  ::  f 

4i=4 
Or,  inverted  f  :  f  : :  |  =  LO  —  9  Ans. 

2.  If  6£  hundred  weight  be  carried  22 ^  miles  for  25f 
dollars,  how  far  may  1  hundred  weight  be  carried  for  the 
same  money?  Ans.  145J  miles. 

3.  If  12  men  can   finish  a  piece  of  work   in  37  f  days; 
how  long  will  it  take  16  men  to  do  the  same  work  ? 

Ans.  281  days. 

4.  A  lends  to  B  lOOf  dollars  for  6|  months;  what  sum 
should  B  lend  to  A  for  3|-  years,  to  requite  his  kindness  ? 

Ans.  14|f  f  dollars. 

5.  Mow  many  feet  long  must  a  board  be,  that  is  |  of  a 
root  wide,  to  equal  one  that  is  20£  feet  long,  and  f  of  a  foot 
wide?  Ans.  17  i  lert  long. 

6.  In  exchanging  20£  yards  of  cloth  of  1  i  yard  wide,  for 
some  of  the  same  quality  of  |  yard  wide,  what  quantity  of 
the  latter  makes  an  equal  barter  ? 

Ans.  34£  yards. 


INVOLUTION. 


12? 


PART  VII. 

EXTRACTION  OF  THE  ROOTS,  AND  COMPARATIVE 
ARITHMETIC. 


SECTION  1. 

Involution,  or  the  Raising  of  Powers. 

INVOLUTION  is  the  multiplying  of  a  given  number  by 
itself  continually,  any  certain 'number  of  times. 

The  product  of  any  number  so  multiplied  into  itself,  is 
termed  the  power  of  that  number. 

Thus  2x2  =  4=  the  second  power  or  square  of  2* 

2x2x2  =  8=  the  third  power  or  cube  of  2. 
2x2x2x2  =  16=  the  fourth  power  of  2,  &c. 

The  number  denoting  the  power  to  which  any  give  n  sum 
is  raised,  is  called  the  index  or  exponent  of  that  powe  r. 

If  two  or  more  powers  are  multiplied  together,  ttu  ir  pro- 
duct will  be  that  power,  whose  index  is  the  sum  of  the  expo- 
nents of  the  factors.  Thus  2x  2=4,  the  2d  powei  of  2  ,• 
4X4=16,  the  4th  power  of  2;  16x16  =  256,  tlie  8th 
power  of  2,  &c. 

TABLE 
Of  the  first  nine  powers. 


1 

CU 

rt 
Jf 

i 

o 

o 

1 

1 

1 
«S 

1 
1 

ft 

o 

1 

00 

•5 

1 

1! 

11 

11     1 

1 

11               1| 

1 

2 

4, 

8| 

16!       32 

64 

128           256| 

512 

3| 

yi 

27| 

81      243 

729|       2187)         6561! 

19683 

4|I6| 

64  j 

256|   1024 

4096|     16384J       65536) 

262144 

5|25|125|  625 

3125|  15625)     78125J'    3J0625| 

1953125 

6|36j 

216|12:)6|  7776, 

46656) 

279936|   167961  6j 

10077696 

7|4i)|343|2401|16807|11764<)| 

823543|  5764801J 

40353607 

8|64|512)4096!32768 

262114|2097152|  16777216J  i  34217728 

[  9(81  72:)|6561I5904;): 

531441  14782)6:)  43046721)3874204891 

123  EVOLUTION.— SQUARE  ROOT. 

EXAMPLE. 

1.  What  is  the  3d  power  of  15  ? 

15x15x15=3375     Ans. 
2    What  is  the  4th  power  of  35  ?  Ans.  1500625. 

3.  What  is  the  third  power  of  1,03  1      Ans.  1,092727. 

4.  What  is  the  5th  power  of  ,029  ? 

Ans.  ,000000707281. 

5.  What  is  the  4th  power  of  J  1  Am.  -fa 


SECTION  2. 

Of  Evolution,  or  the  Extracting  of  Roots. 

EVOLUTION  js  the  reverse  of  involution.  For  as  3x  3=9 
y  3  =  27,  the  power ;  so  27-^-3=9—3=3,  the  root  of  that 
power.  Hence  the  root  of  any  number,  or  power,  is  such 
a  number  as  being  multiplied  into  itself  a  certain  number  of 
times,  will  produce  that  power.  Thus,  4  is  the  square  root 
of  16,  for  4X4=16;  and  5  is  the  cube  root  of  125,  for 
5x  5x  5=125. 


SECTION  3. 
THE  SQUARE  ROOT. 

number  multiplied  once  into  itself  is  called  the 
square  of  that  number.  Hence,  to  extract  the  square  root 
<;f  any  number,  is  to  find  such  a  number  as  being  multiplied 
f>y  itself  will  be  equal  to  the  ^iven  number. 

RUf.R. 

1.  Point  off  the  given  sum  into   periods  of  two  figures 
«  a_,  beginning  at  the  right  hand. 


SQUARE  ROOT. 


12!) 


2.  Subtract  from  the  first  period  on  the  left,  the  greatesi 
square  contained  therein  ;  setting  the  root,  so  found,  for  the 
first  quotient  figure. 

3.  Double  the  quotient  for  a  new  divisor,  and  bring  down 
trte  next  period  to  the  remainder  lor  a  new  dividual.     Try 
how  often  the  divisor  is  contained  in  the  dividual,  omitting 
the  units. figure,  and  place  the  {lumber,  so  found,  in  the  quo- 
tient, and  on  the  right  of  the  divisor  ;  multiply  and  subtract 
as  in  division. 

4.  Double  the  quotient  for  a  new  divisor  ;  bring  down  the 
next  period,  and  proceed  as  before,  till  all  the  periods  are 
brought  down.    When  a  remainder  occurs,  add  ciphers  for  a 
new  period,  the  quotient  figure  of  which  will  be  a  decimal, 
which  may  be  extended  to  any  required  degree  of  exactness. 

PROOF.  * 

Square  the   root,  adding   the   remainder  (if  any)  to  the 
product,  which  will  equal  the  given  number. 

EXAMPLE, 

1.  What  is  the  square  root  of  531441  ? 

53144J      ( 729     An*. 

49 

I.  double  the  quotient  14,2 )     414 

284 


2.  double 


do.    144,9)  13041 
13041 
729 

729 


6561 
1458 
5103 

Ans.  327 
2187. 
6561. 
19683. 

4698. 
6031. 

1506,23  -t 
2756,22^4 

531441  proof. 
2.  What  is  the  square  root  of  106929  ? 
3.  What  is  the  square  root  of  4782969  ? 
4.  What  is  the  square  root  of  43046721  ? 
5.   What  is  the  square  root  of  387420489  ? 
6.  What  is  the  square  root  of  22071204  ? 
7.   What  is  the  square  root  of  36372961  ? 
8,   What  is  the  square  root  of  2268741  ? 
9.   What  is  fhe  square  roof  of  7596796  ' 

130  SQUARK  ROOT. 

When  there  are  decimals  joined  to  the  whole  numbers  in 
/he  given  sum,  make  the  number  of  decimals  even  by  adding 
ciphers,  and  point  off  both  ways,  beginning  at  the 'decimal 
point. 

10.  What  is  the  sqiuire  root  of  9712,718051  ? 

Ans.  98,553  + 

11.  What  is  the  square  root  of  3,1721812  ? 

Ans.  1,78106  + 

12.  What  is  the  square  root  of  4795,25731  ? 

Ans.  69,247 

13.  What  is  the  square  root  of  ,00008836  ? 

Ans.  ,0094 

To  extract  the  square  root  of  a  vulgar  fraction. 

RULE. 

Reduce  the  fraction  to  its  lowest  term  ;  then  extract  the 
"Kjuare  root  of  the  numerator  for  a  new  numerator,  and  the 
scjuare  root  of  the  denominator  for  a  new  denominator. 

Nute.  When  the  fraction  is  a  surd*  that  is,  a  number  whose  exact 
ff>ot  cannot  be  found,  reduce  it  to  u  decimal  and  extract  the  root  there- 
from. 

EXAMPLES, 

1 .   What  is  the  square  root  of  Jf  £A  7  Ans.  J 

"2.   What  is  th<;  square  root  of  |Jf  A  !  f 

3.  What  is  the  square  root  of  iff  f£  ?  ^f 

Surds. 

4.  What  is  the  square  root  of  ffj  '!          Ans.  ,86602  + 

5.  What  is  the  square  root  of  f  Jf  ?  ,93309  + 

6.  What  is  ihr  square  root  of  f|J  ?  ,72414  + 

To  ertraci  thr  vjimre  root  of  a  mired  number. 

RUTJv 

1.  Reduce  the  fractional  part  of  the  mixed  number  to  its 
lowest  term,  and  thn  mixed  number  to  an  improper  fraction. 


SQTTARK  ROOT.  131 

2.  Extract  the  roots  of  the  numerator  and  denominator, 
for  a  new  numerator  and  denominator. 

If  the  mixed  number  given  be  a  surd,  reduce  the  frac- 
tional part  to  a  decimal,  annex  it  to  the  whole  number,  and 
extract  the  square  root  therefrom. 

EXAMPLES. 

1.  What  is  the  square  root  of  37 ^f?  Ans.  6-J 

2.  What  is  the  square  root  of  27  r9F  ?  5i 

3.  What  is  the  square  root  of  51|}?  7} 

4.  What  is  the  square  root  of  9ff  ?  3i 

Surds. 

5.  What  is  the  square  root  of  7T9T?  Ans.  2,7961  + 

6.  What  is  the  square  root  of  8f  ?  2,951 9  + 

7.  What  is  the  square  root  of  85-ff  ?  9,27  -f 


Any  two  sides  of  a  right  angled  triangle  given  to  find  the 
third  side. 


RULE. 

As  the  square  of  the  hypothenuse  or  longest  side,  is  al- 
ways equal  to  the  square  of  the  base  and  perpendicular,  the 
other  sides  added  together ;  then  it  is  plain  if  the  length  of 
the  two  shortest  sides  are  given,  the  square  root  of  both 
these  squared  and  added  together,  will  be  the  length  of  the 
third  or  longest  side. 

Again,  when  the  hypothenuse,  or  longest  side,  and  one  of 
the  others  are  given 5  the  square  root  of  the  difference  of  the 
squares  of  these  two  given  sides  will  be  the  len«  ;th  of  the 
remaining  side. 


132 


SQUARE  ROOT. 


EXAMPLE. 


1.  The  wall  of  a  fortress  is  3fi  feet  high,  and  the  ditch 
before  it  is  27  feet  wide:  it  is  required  to  find  the  length  of 
n  ladder  that  will  reach  to  the  top  of  the  wall  from  tho  op- 
posite side  of  the  ditch  ?  Ans.  45  feet. 


27  ft.  ditch 


2.  The  top  of  a  castle"  from  the  ground  is  45  yards  high, 
and  is  surrounded  with  a  ditch  60  yards  broad,  what  length 
must  a  cable  be  to  reach  from  the  outside  of  the  ditch  to  the 
top  of  the  castb?  Ans.  75  yards. 

8.  In  a  right  angled  triangle,  ABC,  the  hypothenuse 
line  A  C  is  45  feet,  the  base  A  B  27  feet;  required  the  length 
of  the  perpendicular  line  B  C?  Ans.  36  feet. 


B 


4.  In  a  right  angled  triangle,  A  B  C,  the  line  A  C  is  75 
feet,  B  C  45  feet;  required  the  length  of  the  line  A  B? 

Ans.  60  feet. 

To  find  the  side  of  a  square  equal  in  area  to  any  given  .«/• 


RULE. 

Extract  the  square  root  of  the  content  of  the  given  su 
|H;rfices  ;  the  quotient  will  give  the  side  of  the  equal  square 
sought. 


SQUARK  ROOT  133 

EXAMPLES. 

1.  If  the  content  oi  a  given  circle  be  ICO,  what  is  the 
side  of  the  square  equal?  Ans.  12,(>4911-{- 

2.  If  the  area  of  a  circle  he  '2025,  what  is  the  side  of  the 
square  equal  ?  Ans.  45. 

3.  It*  the  area  of  a  circle  be  750,  what  is  the  side  of  the 
square  equal?  Ans.  27,38612  + 

To  find  tht  diameter  of  a  circle  of  a  given  proportion 
larger  or  less  than  a  given  one. 

RULE. 

Square  the  diameter  of  the  given  circle,  and  multiply 
(if  greater)  or  divide  (if  less)  the  product,  by  the  number  of 
times  the  required  circle  is  greater  or  less  than  t.\c  given 


EXAMPLES. 

1.  There  is  a  circle  whose  diameter  is  4  feet;  what  is  the 
diameter  of  one  4  times  as  large  ?  Ans.  8  feet. 

2.  A  has  a  circular  yard  of  100  feet  diameter,  but  wishes 
to  enlarge   it  to  one  of  3  times  that  area;   what  wall  the 
diameter  of  the  enlarged  one  measure?  Ans.  173,2 -f 

3.  If  the  diameter  of  a  circle  be  12  inches,  what  will  be 
the  diameter  of  another  circle  of  half  the  size  ? 

Ans.  8,48  -f  inches. 

The  area  of  a  circle  given  to  find  the  diameter. 
RULE. 

Multiply  the  square  root  of  the  area  by  1,12837,  and  the 
produce  will  be  the  diameter. 

EXAMPLES. 

1.  When  the  area  is  160,  what  is  the  diameter? 

Ans.  1 4,272947 -f 

2.  What  length  of  a  halter  will  be  sufficient  to  fasten  a 
horse  from  a  post  in  the  centre,  so  that  he  may  be  able  to 
graze  upon  an  acre  of  grass,  and  no  more? 

Ans.  7,1364  perches,  or  117  ft.  9  inches* 
M 


134  SQUARE  ROOT. 

*          Application. 

1.  If  an   army   of  20736  men  is  formed  into  a   square 
column ;  how  many  men  will  each  front  contain? 

Ans.  144  men. 

2.  How  many  feet  of  boards  will  it  require  to  lay  the 
floor  of  a  room  that  is  25  feet  square?  Ans.  625  feet. 

3.  A  certain  square  pavement  contains   191736  square 
stones,  all  of  the  same  size ;  how  many  are  contained  in  one 
of  its  sides  ?  Ans.  444. 

4.  In  a  triangular  piece  of  ground  containing  600  perches, 
one  of  the  shortest  sides  measures  40  perches,  and  the  other 
30 ;  what  is  the  length  of  the  longest  side  ? 

Ans.  50  perches. 

5.  Two  gentlemen  set  out  from  Pittsburgh  at  the  same 
time ;  one  of  them  travels  84  miles  due  north,  and  the  other 
50  miles  due  west ;  what  distance  are  they  asunder  ? 

Ans.  97^  -{-  miles. 
ft.   What  is  the  square  root  of  964,5192360241  ? 

Ans.  31,05671. 

7.  What  is  the  square  root  of  1030892198,4001  ? 

Ans.  32107,51. 

As  it  is  probable  many  teachers  rind  it  difficult  to  explain 
satisfactorily  the  reasons  and  principles  upon  which  the  rules 
for  the  extraction  of  the  roots  are  founded,  I  have  subjoined 
the  following  demonstration  of  the  rule  for  extracting  the 
square  root ;  and  which  will  also  serve  to  show  the  reason 
of  the  rules  for  extracting  the  roots  of  the  higher  powers. 
From  what  has  been  already  said  on  this  rule,  it  is  sufficiently 
evident  that  the  extraction  of  the  square  root  has  always  this 
operation  on  numbers,  viz.  to  arrange  the  number  of  which 
the  root  is  extracted  into  a  square  form.  Thus,  if  a  car- 
penter should  have  625  feet  of  dressed  boards  for  laying  a 
floor ;  if  he  extracts  the  square  root  of  this  number,  (625) 
he  will  have  the  exact  length  of  one  side  of  a  square  floor, 
which  these  boards  will  be  sufficient  to  make. 

Let  this  then  be  the  question :  Required  to  find  the  length 
of  one  side  of  a  square  room,  of  which  625  square  feet  of 
boards  will  be  sufficient  to  lay  the  floor. 

The  first  step,  according  to  the  rule  given,  is  to  point  off 
the  numbers  into  periods  of  two  figures  each,  beginning 


at  the  unit's  place.     This  ascertains  the  number  of  figures 
of  which   the  root  will  consist,  from  this  principle,  that  the 
product  of  any  two  numbers  can  have,  at  most,  but  so  many  . 
places  of  figures,  as  there  are  places  in  both  the  factors,  and 
at  least,  but  one  less. 

The  number  (625)  will  then  have  two  periods,  and  con- 
sequently the  root  will  consist  of  two  figures. 

Operation.  The  last,  or  left-hand  period  in  this 

.    .  number  is  6,  in  which  4  is  the  greatest 

625  (  2  square,  and  2  the  root ;  hence  2  is  the 

4  first  figure  in  the  root,  and  as  one  fig- 

ure  more  is  yet  to  be  found,  we  may 

225  for   the   present    supply  the    place  of 

that  figure  with  a  cipher  (20) ;  then  20 
Figure  1.  will  express  the  just  value  of  that  part 

of  the  root  now  obtained.     But  a  root 
is  the  side  of  a  square,  of  equal  sides. 


A 


20 
20 


Hence,   figure    1    exhibits    a    square, 
each  side  of  which  is  20  feet,  and  the 
a|        400    |b  area  400,  of  which  20  is  the  root  now 

20  obtained. 

As  the  rule  requires,  we  next  subtract  the  greatest  square 
contained  in  the  first  period,  and  to  the  remainder  bring 
down  the  next  period.  4  is  the  greatest  square  contained  in 
the  first  period  (6),  and  as  it  falls  in  the  place  of  hundreds, 
is  in  reality  400,  as  may  be  seen  by  filling  up  the  places  to 
the  right-hand  with  ciphers ;  this  subtracted  from  the  (i 
leaves  2,  as  a  remainder,  to  which  if  the  next  period  is 
brought  down,  the  remainder  will  be  225  ;  and  the  original 
number  of  feet  (625)  has  been  diminished  by  the  deduction 
of  400  feet,  a  number  equal  to  the  superficial  content  of  the 
square  A. 

Figure  1,  therefore,  exhibits  the  exact  progress  of  the 
operation,  and  shows  plainly  how  400  feet  of  the  boards 
have  been  "disposed  in  the  operation  thus  far,  and  also  that 
225  feet  yet  remains  to  be  added  to  this  square,  by  en- 
larging it  in  such  a  manner  as  not  to  destroy  its  quadrate 
form,  or  its  continuing  a  complete  and  perfect  square. 
Should  the  addition  be  made  to  one  side,  only,  the  figure 
would  lose  its  square.  The  addition  must  be  made  to  two 
sides:  accordingly  the  rule  directs  to  lt  double  the  quotient 


136 


SQUARE  ROOT. 


(viz.  the  root  already  found)  for  a  new  divisor  ;"  the  double 
of  the  root  is  equal  to  two  sides  of  the  square  A,  for  the 
double  of  2  is  4,  and  as  this  4  falls  in  the  place  of  tens, 
since  the  next  figure  in  the  root,  according  to  the  rule,  is  to 
be  placed  before  it  in  the  place  of  units,  it  is  in  reality  40, 
and  equal  to  a  b  and  b  c  which  are  20  each. 

Operation  continued.  Again,    as   the   rule   di- 

.    .  rects,  try  how  often  the  di~ 

625  ( 25  visor   is   contained   in   the 

4  dividual,  omitting  the  units 

figure. 

45)225  The   divisor   is    here    4, 

225  which,  as  has  already  been 

shown,    is   4   tens,   or   40 ; 
000  this  is   to  be   divided   into 

the  remainder  225 ;  omit- 
n  5  ting  the  last  figure=220. 
But  40— the  sum  of  the 
two  sides  b  c  and  c  d /  to 
which  the  remaining  225 
is  to  be  added,  and  the 
square  A  enlarged,  which 
omitting  the  last  figure  (5) 
gives  5  for  the  last  quo- 
tient figure ;  5  is  the  breadth 
of  the  two  parallelograms 
B  B,  the  area  of  each  (5  X 
20  b  5  20)  is  100. 

The  rule  requires  us  to  omit  the  last  figure  in  the  divid- 
ual^ and  also,  to  place  the  quotient  figure,  when  found,  on 
the  right  of  the  divisor ',  the  reasons  for  which  are,  that  ad- 
ditions of  the  two  parallelograms  B  B  ta  the  sides  of  the 
square  A  (fig.  2)  do  not  leave  it  a  perfect  square,  a  deficiency 
remaining  at  the  corner  D  ;  the  right-hand  figure  is  omitted 
to  leave  something  of  the  dividend  for  this  deficiency. 
And  as  this  deficiency  is  limited  by  the  two  parallelograms 
B  B,  and  the  quotient  figure  (5)  is  the  breadth  of  these, 
consequently  the  quotient  5= the  length  of  each  of  the 
sides  of  the  small  square  D  ;  this  quotient  then  being 
placed  on  the  right  of  the  divisor  and  multiplied  into  itself 
gives  the  area  of  the  square  I)  ;  which  being  added  to  the 


20 

5 

B          5 

D    5 

en 

d 

100 

25 

c 

A 

B  25 

0 

20 

20 

20 

5 

a 

400 

100 

X 

CUR!-:  ROOT.  137 

contents  of  the  two  parallelograms  B  B  each  (100)  200, 
shows  that  the  remaining  225  feet  of  boards  have  kasji  dis- 
posed of,  in  these  three  additions  (B  B  D)  made  to  the  T^st 
square  A ;  whilst  the  figure  is  seen  to  be  continued  a  corrt* 
plete  square. 

Q.  E.  D. 
PROOF. 

The  square  A  =400  feet 

The  parallelograms  B  B  =200 
The  square  D  =   25 

625  feet. 


SECTION  4. 
THE   CUBE   ROOT. 

THE  cube  is  the  third  power  of  any  number,  and  is  found 
by  multiplying  that  number  twice  into  itself.  As  2  X  2  x  2—^. 

To  extract  the  cube  root,  therefore,  of  any  number,  is  to 
find  another  number,  the  cube  of  which  will  equal  the  given 
number.  Thus  4  is  the  cube  root  of  64  ;  for  4X  4x  4=64. 

RULE. 

1.  Point  off  the  given  number  into  periods  of  three  figures 
each,  beginning  at  the  units  place,  or  decimal  point.     These 
periods  will  show  the  number  of  figures  contained  in  the 
required  root. 

2.  Find  the  greatest  cube  contained  in  the  first  period, 
and  subtract  it  therefrom  ;  put  the  root  of  this  cube  in  the 
quotient,  and  bring  down  the  next  period  to  the  remainder 
for  a  new  dividual. 

3.  Square  the  quotient  and  multiply  it  by  3  for  a  defective 
divisor;  2x2x3  =  12.     Find  how  often  this  is  contained 
in  the  dividual,  rejecting  the  units  and  tens  therein,  and 
place  the  result  in  the  quotient,  and  its  square  to  the  right  of 
the  divisor.     4  x  4=16  put  to  the  divisor  12  = 

M2 


138 


CUBE  TtOOT. 


4.  Multiply  the  last  tigure  in  the  quotient  by  the  rest,  and 
the  product  by  30  ;  add  this  to  the  defective  divisor,  ana 
multiply  this  sum  by  the  last  figure  in  the  quotient,  subtract 
that  product  from  the  dividual,  bring  down  the  next  period, 
and  proceed  as  before. 

Note.  When  the  quotient  is  1,  2,  or  3,  put  a  cipher  in  the  place  of 
tens  in  filling  up  the  square  on  the  right  of  the  divisor. 


EXAMPLE. 

U  What  is  the  cube  root  of  48228544  1 


Greatest  cube  in  48  is 


Operation. 

48228544(364  Ans. 
27 


21228 

Square  of  3  X  by  3=27.  1  def.  divis. ' 
Square  of  6  put  to  27  =2736 

6  last  -quo.  fig.  X  by  the  rest 

and  30  =   540 

Complete  divisor  3276  J  19656 


Square  of  36  X  3=3888.  2  def.  divis. 
Square  of  4  put  to  3888  =  388816 
4  last  quo.  fig.  X  by  the  rest 

and  30  =  4320 


Complete  divisor 


393136 


1572544 


1572544 


2.  What  is  the  cube  root  of  13824  T 
3.  What  is  the  cube  root  of  373248  ? 
4.  What  is  the  cube  root  of  5735339  ? 
5.  What  is  the  cube  root  of  84604519  ? 
6.  What  is  the  cube  root  of  27054036008  I 
7.  What  is  the  cube  root  of  122615327232  ? 
8.  What  is  the  cube  root  of  22069810125  ? 
9.  What  is  the  cube  root  of  219365327791  ? 
10.  What  is  the  cube  root  of  673373097125  ? 
11.  What  is  the  cube  root  of  12,977875  ? 
12.  What  is  the  cube  root  of  15926,972504  ? 
13.  What  is  the  cube  root  of  36155,027576  ? 

Ans.  24 
72 

179 
439 
3002 
4968 
2805 
6031 
8765 
2,35 
?Mfi  f 
33,06  -f 

CUBE  ROOT.  139 

14.  What  is  the  cube  root  of  ,053258279  ?  Ans.  ,376  + 

15.  What  is  the  cube  root  of  ,001906624  ?  ,124 

16.  What  is  the  cube  root  of  ,000000729  ?  ,009 

17.  What  is  the  cube  root  of  2  ?  1,25  + 

To  extract  the  cube  root  of  a  vulgar  fraction. 

RULE. 

Reduce  the  fraction  to  its  lowest  terms  ;  then  extract  the 
cube  root  of  the  numerator  for  a  new  numerator,  and  the 
cube  root  of  the  denominator  for  a  new  denominator ;  but 
if  the  fraction  be  a  surd,  reduce  it  to  a  decimal,  and  extract 
the  root  from  it  for  the  answer. 

EXAMPLES. 

1.  What  is  the  cube  root  of  f  f-f  1  Ans.  4 

2.  What  is  the  cube  root  of  T3/^  ?  f 

3.  What  is  the  cube  root  of  4-f  f  £  ?  f 

Surds. 

4.  What  is  the  cube  root  of  4  ?  Ans.  ,829 -f- 

5.  What  is  the  cube  root  of  f  ?  ,873 -f 

6.  What  is  the  cube  root  of  f  ?  ,822 -f 

To  extract  the  cube  root  of  a  mixed  number. 

RULE. 

Reduce  the  fractional  part  to  its  lowest  terms,  and  the 
mixed  number  to  an  improper  fraction  ;  extract  the  cube 
roots  of  the  numerator  and  denominator  for  a  new  numerator 
and  denominator  ;  but  if  the  mixed  number  given  be  a  surd, 
reduce  the  fractional  part  to  a  decimal,  annex  it  to  the  whole 
number,  and  extract  the  root  therefrom. 

^  EXAMPLES. 

1.  What  is  the  cube  root  of  31  J/j  ?  Ans.  3| 

2.  What  is  the  cube  root  of  12if  ?  2J 

3.  What  is  the  cube  root  of  405r2^  ?  7f 


140  CUBE  ROOT. 

Surds. 

4.  What  is  the  cube  root  of  7}  1  Ans.     1,93-f- 

5.  What  is  the  cube  root  of  8f  ?  2,057  + 

6.  What  is  the  cube  root  of  9£  ?  2,092  + 

To  find  the  side  of  a  cube  that  shall  be  equal  to  any  given 
solid,  as  a  globe,  a  cone,  <fyc. 

RULE. 

Extract  the  cube  root  of  the  solid  content  of  any  solid 
body,  for  the  side  of  the  cube  of  equal  solidity. 

EXAMPLES.  * 

1.  If  the  solid  content  of  a  globe  is  10648,  what  is  the 
side  of  a  cube  of  equal  solidity  ?  Ans.  22. 

2.  If  the  solid  content  of  a  globe  is  389017,  wha't  is  the 
side  of  a  cube  of  equal  solidity?  Ans.  73. 

Note.  The  relative  size  of  different  cubical  vessels  is  found  by  mul- 
tiplying the  cube  oi'  the  side  of  the  given  vessel,  by  the  proportional 
number,  and  taking  the  cube  root  of  the  product  for  the  answer  sought. 

EXAMPLES. 

1,  There  is  a  cubical  vessel  whose  side  is  two  feet  •  I  de- 
mand the  size  of  another  vessel  which  shall  contain  three 
times  as  much  1  Ans.  2  feet  10  inches  and  }  nearly. 

2.  There  is  a  cubical  vessel  whose  side  is  1  foot ;  re- 
quired  the  side  of  another  vessel  that  shall  contain  three 
times  as  much  ?  Ans.  17,306  inches 

Application. 

1.  If  a  ball  of  6  inches  diameter  weigh  32  lb.,  what  will 
one  of  the  same  metal  weigh,  whose  diameter  is  3  inches  ? 

Ans.  4  lb. 

2.  What  is  the  side  of  a  cubical  mound  equal  to  one  28H 
feet  long,  216  broad,  and  48  high?  An*.   144  feet. 

3.  There  is  a  stone  of  cubic  form,  which  contains  389017 
solid  feet ;  what  is  the  superficial  content  of  one  of  its  sides  ? 

AIM.  5329  feet. 


PROGRESSION.  141 

4.  What  is  the  difference  between  half  a  solid  foot,  and  a 
solid  half  foot  ?  Ans.  3  half  feet. 

5.  In  a  cubical  foot,  how  many  cubes  of  6  inches,  and 
how  many  of  4  are  contained  therein  ? 

Ans.  8  of  6  inches,  and  27  of  4  inches. 


SECTION  5. 
OF  PROGRESSION. 

PROGRESSION  is  of  two  kinds,  arithmetical  and  geometrical. 

Arithmetical  progression  is  when  any  series  of  numbers 
increase  or  decrease  regularly  by  a  common  difference.  As 
1,  2,  3,  4,  5,  6,  &c.  are  in  arithmetical  progression  by  the 
continual  adding  of  one ;  and  9,  7,  5,  3,  1,  by  the  continual 
subtracting  of  two. 

Note. — In  any  series  of  even  numbers  in  arithmetical  progression, 
the  sum  of  the  two  extremes  will  be  equal  to  the  sum  of  any  two  terms, 
equally  distant  therefrom  ;  as  2.  4.  6.  8.  10.  12.,  where  2  -f- 12  =  14,  so 
4  -f 10  =  14,  and  6  -f  8  =  14.  But  if  the  number  of  terms  is  odd,  the 
double  of  the  middle  term  will  be  equal  to  any  two  of  the  terms 
equally  distant  therefrom ;  as  3.  6.  9.  12.  15.  where  the  double  of  9  the 
middle  term  =  18,  and  3 -f  15  =  18,  or  64-12  =  18. 

In  arithmetical  progression  five  things  must  be  carefully 
observed,  viz. 

1.  The  first  term, 

2.  The  last  term, 

8.  The  number  of  terms, 

4.  The  equal  difference, 

5,  The  sum  of  all  the  terms." 

Casel. 

The  first  term,  common  difference,  and  number  of  terms, 
given  to  find  the  last  term,  and  sum  of  all  the  terms. 

RULE. 
1 .  Multiply  the  number  of  terms,  less  1 ,  by  the  common 


142  PROGRESSION. 

difference,  and  to  the  product  add  the  first  term,  the  sum 
will  be  the  last  term. 

2.  Multiply  the  sum  of  the  two  extremes  by  the  number 
of  terms,  and  half  the  product  will  be  the  sum  of  all  the 
terms. 


1.  A  merchant  bought  50  yards  of  linen,  at  2  cents  for 
the  first  yard,  4  for  the  second,  6  for  the  third,  &c.  increas- 
ing two  cents  every  yard ;  what  was  the  price  of  the  last 
yard,  how  much  the  whole  amount,  and  what  the  average 
price  per  yard  ? 

50  number  of  terms 
1 

49  number  of  terms  less  one 
Multiply  by         2  common  difference 

98 
Add  2  first  term 

100  last  term 

2  +  100=102  sum  of  the  two  extremes 
Multiply  by       50  number  of  terms 


2)5100 


50 )  25,50  sum  of  all  the  terms 


51  cents 

C      100  cents  the  last  yard 
Ans.    <  25,50   do.    the  whole  amount 

f        51    do.    the  average  price  per  yd. 

2.  Bought  20  yards  of  calico  at  3  cents  for  the  first  yard, 
6  for  the  second,  9  for  the  third,  &c. ;  what  did  the  whole 
cost?  Ans.  6  dolls.  30  cents. 

3.  If  100  apples  were  laid  two  yards  distant  from  each 
"other,  in  a  right  line,  and  a  basket  placed  two  yards  distant 
from  the  first  apple,  what  distance  must  a  person  travel  to 
gather  them  singly  into  the  basket  ? 

Ans.  11  miles,  3  furlongs,  180  vardau 


PROGRESSION.  1  13 

4.  A  agreed  to  serve  B  10  years,  at  the  rate  of  20  dol- 
lars for  the  first  year,  30  for  the  second,  40  for  the  third, 
&c.  ;  what   had   he  the  last  year,  how  much  for  the  whole 
time,  and  what  per  annum  'I 

Ans.  110  dolls,  for  the  last  year,  650  dolls,  the  whole 
amount,  and  65  dolls,  per^xnnum. 

5.  A  sold  to  B  1000  acres  of  land,  at  10  cents  for  the 
first  acre,  20  for  the  second,  30  for  the  third,  &c. ;  what  was 
the^price  of  the  last  acre,  and  what  did  the  whole  come  to? 

j         $       100  dolls,  the  last  acre, 
'    I  50050    do.     whole  cost. 

Case  2. 

When  the  two  extremes,  and  number  of  terms  are  given, 
td  find  the  common  difference. 

RULE. 

Divide  the  difference  of  the  extremes,  by  the  number  of 
terms,  less  one  ;  the  quotient  will  be  the  common  difference. 

EXAMPLE. 

1 .  A  is  to  receive  from  B  a  certain  sum  to  be  paid  in  1 1 
several  payments  in  arithmetical  progression  ;  the  first  pay- 
ment to  be  20  dollars,  and  the  last  to  be  100  dollars  ;  what 
is  the  common  difference,  what  was  each  payment,  and  how 
much  the  whole  debt  ? 

Operation, 

100  last  term 
20  first  term 

No.  of  terms  11 — 1  =  10)80  the  difference 

8  common  difference 
20 -f  100X  5i=660  whole  debt 

20  first  payment 
20  +  8=28  second  do. 
28 -j- 8 =36  third     do,     &c. 

2.  There  are  21  persons  whose  ages  are  equally  distant 
from  each  other,  in  arithmetical  progression ;  the  youngea! 


1 44  GEOMETRIC  A  L  PROGRESSION. 

ts  20  years  old,  and  the  eldest  60  ;  \\  hat  is  the  common  dif- 
ference of  their  ages,  and  the  age  of  each  man  ? 

C    2  common  difference 
Ans.  <  20  +  2  —  22  the  second 

(22  4-2  =  24  the  third,  &c. 

3.  A  man  is  to  travel  from  Pittsburgh  to  a  certain  place, 
tn  12  days,  and  to  go  but  three  miles  the  first  day,  increas- 
ing each  day's  journey  in  arithmetical  progression,  making 
the  last  day's  travelling  58  miles ;  what  is  the  daily  increase, 
and  what  the  whole  distance  I 

*        (      5  miles  daily  increase 
*  <  366  miles  whole  distance. 


SECTION  6, 

GEOMETRICAL  PROGRESSION. 

ANY  series  of  numbers  increasing  or  decreasing  by  one 
continual  multiplier,  or  divisor,  called  the  ratio,  is  termed 
geometrical  progression;  as  2,  4,  8,  16,  32,  &c.  increase 
by  the  multiplier  2  ;  and  32,  16,  8,  4,  2,  decrease,  continu- 
ally, by  the  divisor  2. 

In  geometrical  progression  there  are  five  things  to  bo 
carefully  observed. 

1.  The  first  term, 

2.  The  last  term, 

3.  The  number  of  terms, 

4.  The  ratio, 

5.  The  sum  of  all  the  terms. 

To  find  the  last  term,  and  sum  of  all  the  series  in  gcf>. 
netricnl  progression,  worh  by  the  folloiring 

RULE. 

1 .  R  iise  the  ratio  in  the  given  sum,  to  that  power  whose 
index  5  hall  always  bo  one  less  than  the  number  of  term? 
give  n  ;  multiply  the  number  so  found  by  the  first  term,  and 
Tho  jrviduct  will  be  the  last  term,  or  greater  extreme. 


GEOMETRICAL  PROGRESSION.  145 

2.  Multiply  the  last  term  by  the  ratio,  from  that  product 
subtract  the  first  term,  and  divide  the  remainder  by  the 
ratio,  less  one ;  the  amount  will  be  the  sum  of  the  series,  or 
of  all  the  terms. 

EXAMPLE. 

1.  Suppose  20  yards  of  broadcloth  was  sold  at  4  mills 
for  the  first  yard,  12  for  the  second,  36  for  the  third,  &c. 
what  did  the  cloth  come  to,  and  what  was  gained  by  the 
sale,  supposing  the  prime  cost  to  have  been  $15  per  yard  ? 

Note.  In  this  question  observe,  the  first  term  is  4,  the  ratio  3,  arid 
the  number  of  terms  20,  consequently  the  ratio  3  must  be  raised  to 
20— 1= 19th  power.  Thus, 

g  S          S         1 

!          I         1       | 

&       3       8      I 

Indices     1234 

Ratio        3          9         27         81 

.        81 

81 

648 

6561= the  8th  power 
6561 

6561 
39366 
32805 
39366 

43046721=  16th  power 
27=  3d  power 


301327047 
86093442 

1162261467=19th  power 
X  4  first  term 

4649045868=20th  or  last  term 
X3 


13947137604 

— 4  first  term 


Ratio  3—1=2)13947137600 


69 73568,80,0=  sum  of  the  series,  or  number  of 
First  cost  of  the  cloth        300,00         mills  for  which  the  cloth  was  sold 

0=gam, 

N 


146  GEOMETRICAL  PROGRESSION. 

2.  A  father  gave  his  daughter  who  was  married  on  the 
first  day  of  January,  one  dollar  towards  her  portion,  prom- 
ising to  double  it  on  the  first  day  of  every  month  for  one 
year  ;  what  was  the  amount  of  her  whole  portion  'I 

Ans.  4095  dollars. 

3.  A  merchant  sold  15  yards  of  satin ;  the  first  yard  for 
is.  the  second  for  2s.  the  third  for  4s.  &c.  in  geometrical 
progression ;  what  was  the  price  of  the  15  yards? 

Ans.  1638Z.  7s. 

4.  A  goldsmith  sold  1  pound  of  gold  at  1  cent  for  the  first 
ounce,  4  for  the  second,  16  for  the  third,  &c. ;  what  did  it 
come  to,  and  what  did  he  gain,  supposing  he  gave  20  dollars 
per  ounce  ? 

Ans.  He  sold  it  for  55924  dollars  5  cents,  and  gained 
55684  dollars  5  cents. 

5.  What  sum  would  purchase  a  horse  with  4  shoes  and 
8  nails  in  each  shoe,  at  one  mill  for  the  first  nail,  2  mills  foi 
the  second,  4  for  the  third,  &c.  doubling  in  geometrical  pro- 
gression to  the  last  ? 

Ans.  4294967  dollars  29  cents  5  mills. 

t>.  What  sum  would  purchase  the  same  horse,  with  the 
same  number  of  shoes,  and  nails,  at  1  mill  for  the  first  nail, 

3  for  the  second,  9  for  the  third,  &c.,  in  a  triple  ratio  of 
geometrical  progression  to  the  last  ? 

Ans.  926510094425  dollars  92  cents. 

7.  What  sum  would  purchase  the  same  horse,  with  the 
same  number  of  shoes,  and  nails,  at  1  mill  for  the  first  nail, 

4  for  the  second,  16  for  the  3d,  &c.,  in  a  quadruple  ratio  of 
geometrical  progression  to  the  last  ? 

Ans.  6148914691236517  dollars  20  cents  5  mills. 

S.  Sold  30  yards  of  silk  velvet,  at  2  pins  for  the  first 
yard,  6  for  the  second,  18  for  the  third,  &c.,  and  these  dis- 
posed of  at  100U  for  a  farthing  ;  what  did  the  velvet  amount 
to,  and  what  was  gained  by  the  sale,  supposing  the  prime 
cost  to  hav<3  IKJCII  100/.  per  yard  '! 

Amount  214469929/.  5s.  3 


An*'   *  Gained    214466929/.  5*.  Sjd. 


SINGLE  POSITION.  147 

SECTION  7. 

OF  POSITION. 

POSITION  is  a  rule  for  finding  the  true  number,  by  one  or 
more  false  or  supposed  numbers,  taken  at  pleasure. 

It  is  of  two  kinds,  viz.  Single  and  Double. 

Single  position  teaches  to  resolve  such  questions  as  require 
but  one  supposed  number. 

RULE. 

1.  Suppose  any  number  whatever,  and  work  in  the  same 
manner  with  it  as  is  required  to  be  performed  in  the  given 
question. 

2.  Then,  as  the  amount  of  the  errors,  is  to  theN 
sum,  so  is  the  given  number  to  the  one  required. 

PROOF. 

Add  the  several  parts  of  the  result  together,  and  if  k 
agrees  with  the  given  sum,  it  is  right. 

EXAMPLES. 

1.  A  schoolmaster  being  asked  how  many  scholars  he  had, 
said,  if  I  had  as  many,  half  as  many,  and  one  quarter  as 
many  more,  I  should  have  1 32  ;  how  many  had  he  ? 

Suppose  he  had     40  As  110:     40     ::     132    :    48  Ana. 

as  many  40  proof  ^ 

§  as  many  20  40 

\  as  many  10  g. 

US  J? 

132 

2.  It  is  required  to  divide  a  certain  sum  of  money  among 
4  persons,  in  such  a  manner  that  the  first  shall  have  £,  the 
second  ^,  the  third  -J,  and  the  fourth  the  remainder,  which 
is  28  dollars  ;  what  was  the  sum  ? 

Suppose  72  As  18   :  72  ::    28    :    112  dolls.  AIM. 

i  is  24  Proof   112 

i  is  18  ,  .    rzr 

1  i«  19  *  ls  d'» 

*  1S  _  i  is  28 

.54  J-  is  18f 

Rem.  18  84 

28  last  share. 


148  DOUBLE  POSITION. 

3.  A,  B,  and  C,  buy  a  carriage  for  340  dollars,  of  which 
A  pays  three  times  as  much  as  B,  and  B  four  times  as  much 
as  C  ;  what  did  each  pay  ?  f  A  paid  240  dolls. 

Ans.  IE  80 

(C  20 

4.  What  is  the  sum  of  which  1,  ]-,  and  £,  make  148  dol- 
lars? Ans.  240  dollars. 

5.  A  person  having  spent  ^,  and  J  of  his  money :  had  26f 
dollars  left;  what  had  he  at  first?  Ans.  100  dollars. 

6.  A,  B,  and  C,  talking  of  their  ages,  B  said  his  age  was 
once  and  a  half  the  age  of  A ;  C  said  his  was  twice  and  T\ 
the  age  of  both,  and  that  the  sum  of  their  ages  was  93  ;  what 
was  the  age  of  *  each?  i  A's  age  12  years. 

Ans.  <  B's          18 
( C's          63 

7.  Seven-eighths  of  a  certain  number  exceeds  four-fifths 
by  6;  what  is  that  number?  Ans.  80. 

8.  A  gentleman  bought  a  chaise,  horse,  and  harness,  for 
360  dollars ;  the  horse  came  to  twice  the  price  of  the  har- 
ness, and  the  chaise  to  twice  the  price  of  the  horse  and  har- 
ness together ;  what  did  he  give  for  each  ? 

(    80  dollars  for  the  horse 
Ans.  <    40  harness 

( 240  chaise 

9.  A  gentleman  being  asked  the  price  of  his  carriage,  an- 
swered that  i,  i,  i,  and  £  of  its  price  was  228  dollars ;  what 
was  the  price  of  the  carriage  ?  Ans.  240  dolls. 

10.  A  saves  £  of  his  wages,  but  B,  who  has  the  same  sal- 
ary? by  spending  twice  as  much  as  A,  sinks  50  dollars  a 
year;  what  is  their  annual  salary?     Ans.  150  dolls,  each. 


SECTION  8. 

DOUBLE  POSITION. 

DOUBLE  POSITION  is  making  use  of  two  supposed  num- 
bers, to  find  the  true  one. 

RULE. 

1.  Take  any  two  numbers,  and  proceed  with  them  ac- 
cording to  the  conditions  of  the  question,   noting  the  errors 


DOUBLE  POSITION.  149 

of  the  results ;  multiply  these  errors  cross-wise,  viz.  the 
first  position  by  the  last  error,  and  the  last  position  by  the 
first  error. 

2.  If  the  errors  be  alike,  that  is,  both  greater,  or  both  less 
than  the  given  number,  take  their  difference  for  a  divisor, 
and  the  difference  of  the  products  for  a  dividend :  but  if  they 
are  unlike,  take  their  sum  for  a  divisor,  and  the  sum  of  the  pro- 
ducts for  a  dividend,  the  quotient  will  be  the  answer  required 

EXAMPLE. 

1 .  A  father  leaves  his  estate  to  be  divided  among  his  three 
sons,  A,  B,  and  C,  in  the  following  manner,  viz.  A  is  to 
have  one-half  wanting  50  dollars,  B  one-third,  and  C  10  dol- 
lars less  than  B ;  what  was  the  sum  left,  and  what  was  each 
son's  share? 

Operation. 

1st.    Suppose  240  dollars. 
Then  240 -f-  2—50=70  A's  part 
240-+-   3=          80  B's  part 
B's  share     80 — 10=         70  C's  part 


Sum  of  all  their  parts  220 


20  er.  too  little, 
2d.    Suppose  300  dollars. 
Then300-f-  2— 50=100  A's  part 
300-:-   3=          100  B's  part 
B's  share  100—10=  90  C's  part 

Sum  of  all  their  parts     290 

10  er.  too  little, 
errors. 
1st.  sup.  240^20=6000 

2d.  sup.  300-^-10=2400 


10)3600(360  An*. 
Proof    360-;-   2 — 50=130 
360-^   3=          120 
120 — 10=          110 

360 

N2 


150  DOUBLE  POSITION. 

2.  A  and  B  have  the  same  income ;  A  saves  the  £  of  his, 
but  B  by  spending  30  dollars  per  annum  more  than  A,  at  the 
end  of  8  years  finds  himself  40  dollars  in  debt ;  what  is  their 
income,  and  how  much  does  each  spend  per  annum  ? 

(  Their  income  is  200  dolls,  per  annum 
Ans.  <  A  spends  175 

(  B  spends  205 

3.  A,  B,  and  C,  would  divide  100  dollars  between  them, 
so  as  B  may  have  3  dollars  more  than  A,  and  C  4  dollars 
more  than  B ;  how  many  dollars  must  each  have  ? 

(  A  30  dollars. 
Ans.  I  B  33 
(C37 

4.  A,  B,  and  C,  built  a  house  which  cost  10,000  dollars; 
A  paid  a  certain  sum,  B  paid  1000  dollars  more  than  A,  and 
C  paid  as  much  as  both  A  and  B ;  how  much  did  each  one 
pay? 

(  A  paid  2000  dolls. 
Ans.  1  B          3000 
I  C          5000 

5.  A  gentleman  has  2  horses  and  a  saddle  worth  50  dol- 
lars, which  saddle  if  he  put  on  the  back  of  the  first  horse, 
will  make  his  value  double  that  of  the  second;  but  if  he  put 
it  on  the  second  horse,  it  will  make  his  value  triple  that  of 
the  first ;  what  is  the  value  of  each  horse  ? 

A       $  First  horse  30  dolls. 
s'  I  Second  do.  40 

6.  The  head  of  a  fish  is  9  inches  long,  and  its  tail  is  as 
long  as  its  head  and  half  its  body,  and  its  body  is  as  long  as 
its  head  and  tail  together ;  what  is  its  whole  length  ? 

Ans.  6  feet. 

7.  A  laborer  hired  40  days  upon  this  condition,  that  he 
should. receive  20  cents  for  every  day  he  wrought,  and  for- 
l<'it  10  cents  for  every  day  he  was  idle;  at  settlement  he  re- 
ceived 5  dollars ;  how  many  days  did  he  work,  and  how 
many  was  he  idle?  Ans.  Wrought  30  days,  idle  10 

8.  A  and  B  vested  equal  sums  in  trade ;  A  gained  a  sum 
equal  to  J  of  his  stock,  and  B  lost  225  dollars ;  then  A's 
money  was  double  that  of  B's ;  what  sum  had  each  vested  ? 

Ans.  600  dollars. 

9.  Divide  15  into  two  such  parts,  so  that  when  the  greater 


PERMUTATION.  151 

is  multiplied  by  4,  and  the  less  by  16,  the  products  will  be 
equal?  Ans.  Greater  12,  less  3. 

10.  A  person  being  asked  in  the  afternoon  what  o'clock 
it  was,  answered,  that  the  time  past  from  noon  was  equal  to 
fa  of  the  time  to  midnight ;  what  o'clock  was  it  ? 

Ans.  36  minutes  past  .1  o'clock. 


SECTION  9. 
PERMUTATION. 

PERMUTATION  is  the  finding  how  many  different  ways  any 
given  number  of  things  may  be  varied  in  position,  or  ar- 
rangement.    Thus,     123     are   six  different  arrange- 
132     ments  made  upon  the  three 
213     figures     1      2     3. 

2  3     1 

3  1     2 
321 

RULE. 

Multiply  all  the  terms  from  an  unit  up  to  the  given  num- 
ber into  one  another,  and  the  last  product  will  be  the  num- 
ber of  changes  required. 

EXAMPLE. 

1.  In  how  many  different  positions  may  5  persons  be 
placed  at  a  table. 

1X2X3X4X5^120  Ans. 

2  How  many  changes  may  be  rung  on  12  bells,  and  how 
Jong  would  they  be  ringing  but  once  over,  allowing  10 
changes  to  be  rung  in  1  minute,  and  the  year  to  contain  365 
days  6  hours  ? 

Ans.  479001600  changes,  and  would  require  91  years, 

3  weeks,  5  days,  and  6  hours. 

3-  Seven  men  not  agreeing  with  the  owner  of  a  boarding 
house  about  the  price  of  boarding,  offer  to  give  100  dollars 
each,  for  as  long  time  as  they  can  seat  themselves  every  day 
differently  at  dinner  ;  this  offer  being  accepted,  how  long 
may  they  stay  ?  Ans.  5040  days,  or  13  years,  295  days. 


152  COMBINATION. 

4.  What  number  of  variations  will  the  9  digits  admit  of? 

Ans.  362880. 

5.  How  many  changes  may  be  made  on  the  26  letters  of 
the  alphabet  ? 

Ans.  403,291,461,126,605,635,584,000,000. 

Quatril.  Trilns.     Billions.     Millions.      Units. 

Note. — From  the  answer  to  this  last  question,  which  amounts  to  a 
number,  of  which  we  cannot  form  any  conceivable  idea,  we  may  dis- 
cover the  surprising  power  of  numbers,  and  also  the  endless  variety  of 
ideas  that  may  be  distinctly  communicated  by  these  26  simple  charac- 
ters. It  will  also  be  evident  from  the  method  of  notation  here  used, 
that  a  row  of  figures  of  any  given  length  whatever,  may  be  numerated, 
though  we  may  be  entirely  unable  to  comprehend  the  amount. 


SECTION  10. 

COMBINATION. 

COMBINATION  of  quantities,  is  the  showing  how  often  a 
less  number  of  things  can  be  taken  out  of  a  greater,  and 
combined  or  joined  tdgether  differently. 

RULE. 

Take  a  series  of  1,  2,  3,  &c.  up  to  the  number  to  be  com- 
bined ;  take  another  series  of  as  many  places,  decreasing  by 
unity  from  the  number  out  of  which  the  combinations  are  to 
be  made ;  multiply  the  first  continually  for  a  divisor,  and  the 
latter  fcr  a  dividend,  and  the  quotient  will  be  the  answer. 

EXAMPLE. 

1.  How  many  combinations  may  be  made  of  7  dollars 
out  of  12? 

1x2x3,  &c.  up  to  7  =5040  divisor. 
Again,  12  the  whole  number  of  terms  less  7  =  5 
Hence  12  x  11  X  10,  &c.  down  to  5  =  3991680  dividend. 
And  5040  )  3991680  (  792    Ans. 

2.  How  many  combinations  can  be  made  of  6  letters  out 
of  24  of  the  alphabet  ?  Ans.   134596. 

3.  In  how  many  different  ways   may  an   officer  select  8 
men  out  of  30,  so  as  not  to  make  the  same  selection  twice 

Ans.  5H52925 


ADDITION  OF  DUODECIMALS.  153 

PART  VIII. 

MENSURATION. 


SECTION  L 

Duodecimals,  or  Cross  Multiplication. 

DUODECIMALS  are  fractions  of  a  foot,  or  of  an  inch,  or 
parts  of  an  inch,  having  12  for  their  denominator.  Inches 
and  parts  are  sometimes  called  primes  ('),  seconds  ("), 
thirds  ('"),  &c. 

The  denominations  are, 
12  Fourths  ("")  make         1  Third 
12  Thirds  -  1  Second 

12  Seconds  1  Inch  in. 

12  Inches  1  Foot  ft. 

Nole. — This  rule  is  much  used  in  measuring1  and  computing  the  di- 
mensions of  the  several  parts  of  buildings ;  it  is  likewise  used  to  find 
the  tonnage  of  ships,  and  the  contents  of  bales,  cases,  boxes,  &c. 

ADDITION  OF  DUODECIMALS. 
RULE. 

Add  as  in  compound  addition,  carrying  1  for  each  12  u. 
the  next  denomination. 

EXAMPLE. 

Ft.    in.   ""  '"  ""  Ft.     in.    "   

25  9  3  5  8  244  6  3  10  5 

34  3  9  2  7  355  9851 

28  10  4  8  4  559  10  9  5  8 

64  11   9  7  2  129  5569 

82  7  5  6  8  895  1  10  5  11 

15  3  7  9  10  651   1759 

44  6  11  2  8  555  9  8  5-  5 

22  3  6  1  5  388  11  10  10  9 


318  8  9  8  4 


154         SUBTRACTION  OF  DUODECIMALS. 

•SUBTRACTION  OF  DUODECIMALS. 
RULE. 

Work  as  in  compound  subtraction,  borrowing  12  when 
accessary* 

EXAMPLE. 

Ft.    in.    '  Ft.     in.    "    '"    "" 

From  125     4     3     8     2  2756     5780 

Take     68     9     2  10     1  1839     9     5  11   10 


Rem.     56     7     0  10     1 


3.  From  a  board  measuring  35  feet,  9  inches,  2  seconds, 
cut  24  feet,  10  inches,  5  seconds,  and  4  thirds;  what  is  left  ? 

Ans.  Wft.  lOin.  Ssec.  8'" 

4.  A  joiner  having  lined  several  rooms  very  curiously 
with   costly   materials,   finds  the  amount   to  be,   in  square 
measure,  803  feet,  Z  inches,  4  seconds  ;  but  several  deduc- 
tions being  to  be  made  for  windows,  arches,  &c.  those  de- 
ductions amounted  to  70  feet,  3  inches,  7  seconds,  10  thirds, 
5  fourths  ;  how  many  feet  of  workmanship  must  he  be  paid 
for?  Ans.  132ft.  llin.  8"  I"'  7"" 

MULTIPLICATION  OF  DUODECIMALS. 

Case  1. 
When  the  feet  of  the  multiplier  do  not  exceed  12. 

RULE. 

Set  the  feet,  or  the  highest  denomination  of  the  multiplier 
under  the  lowest  denomination  of  the  multiplicand,  and  mul- 
tiply as  in  compound  numbers,  carrying  1  for  every  12  from 
;m<i  denomination  to  another,  and  place  the  result  of  the 
;owcst  denomination  in  the  multiplicand  under  its  multiplier. 

TABLE. 


________ 

2.  Feet  multiplied  by  inches,  give  inches, 


1.  Feet  multiplied  by  feet,  give  feet. 

2.  Feet  multiplied  by  inches,  give  inc«^>. 

3.  Feet  multiplied  by  seconds,  give  seconds,  &c. 
4«  Inches  multiplied  by  inches,  give  seconds. 


MULTIPLICATION  OF  DUODECIMALS.  155 

5.  Inches  multiplied  by  seconds,  give  thirds,  &c. 

6.  Seconds  multiplied  by  seconds,  give  fourths. 

7.  Seconds  multiplied  by  thirds,  give  fifths,  &c. 

PROOF. 

Reduce  the  given  sum  to  a  decimal,  or  work  by  the  rules 
of  practice. 


1.  Multiply 


EXAMPLES. 

Ft.  in.    "  Ft.  in. 

8     6     9     by    7     3 

7     3 


Ans. 

Ft 

59 
2 

11     3 

1     8 

3 

.  62 

0  11 

3 

Or  practice. 
3  is  |i|  8     6     9 

7 

3 

59 
2 

11 
1 

3 

8 

3 

62 

0  11 

3  A?is. 

7     3 


Proof  decimally. 
6     9  =  8,5625 
=     7,25 


Ft.  in. 

2.  Multiply  9     5 

3.  7   10 

4.  846 


428125 
171250 
599375 
62,078125 

12 

0,937500 
12 

11,250000 
12 

3,000000 

Ft.  in.  " 

Ans.  36  10  7 

69  10  2 

21    10  5 


Ft.  in. 
by  3  11 

by  8  11 

by  2     74 

5.  What  is  the  price  of  a  marble  slab,  whose  length  is  5 
feet  7  inches,  and  breadth  1  foot  10  inches,  at  1  dollar  and     I 
50  cents  per  foot?  Ans.  15  dolls.  35^  cts. 

6.  There  is  a  house  with  three  tiers  of  windows,  3  in  a 
tier,  the  height  of  the  first  tier  is  7  feet   10  inches,  of  the 
second  6  feet  8  inches,  and  of  the  third  5  feet  4  inches,  and 
the  breadth  of  each  window  is  ^  ft.  11  inches. ;  what  will  the 
glazing  come  to  at  14  cts.  per  ft.  ?     Ans.  32  dolls.  62£  cts.  . 


156 


MULTIPLICATION  OF  DUODECIMALS. 


Case  2. 

When  the  feet  of  the  multiplier  exceeds  12. 
RULE. 

Multiply  by  the  feet  in  the  multiplier,  and  take  parts  for 
the  inches. 

EXAMPLES. 

1.  Multiply  84  feet  6  inches,  by  36  feet  7  inches  and  6 
seconds. 

Operation. 

84     6 


1 

6*cc. 


507     0 
6 

3042     0 
42     3 

3     6 


Ans.  3094     9 
Ft.  in.         Ft.  in. 

2.  Multiply     76     7  by     19  10 

3.  127     6  by     92     4 

4.  184     8  by  127     6 

Ft.    in.  sec. 

5.  Multiply          311 


4     7     by     36     7 
6x6=36 


Ft.    in.  sec. 
Ans.  1518  10  10 
11772     6 
23545 
Ft.  in.  sec. 
5 


(tin.  is 


II 


1868     3     0 
6 

11209     9     0     " 
155     8      3     6 
25       9 


8      7 


7   ' 
10    4 


21      987 


11402      0      (I 


1  I 


MULTIPLICATION  OF  DUODECIMALS.  157 

6.  A  floor  is  70  feet  8  inches,  by  38  feet  11  inches ;  how 
many  square  feet  are  therein  1 

Arts.  2750ft.  lin.  4sec. 

7.  If  a  ceiling  be  59  feet  9  inches  long,  and  24  feet  6 
inches  broad,  how  many  yards  does  it  contain  ? 

Arts.  I62yds.  5ft.  W$in. 

Note.  Divide  the  square  feet  by  9,  and  the  quotient  will  be  square 
yards. 

8.  What  will  the  paving  of  a  court  yard  come  to  at  15 
cents  per  yard,  the  length  being  58  feet  6  inches,  and  the 
breadth  54  feet,  9  inches  ? 

Arts.  53  dolls.  38  +  cts. 

9.  What  is  the  solid  content  of  a  bale  of  goods,  measur- 
ing in  length  7  feet  6  inches,  breadth  3  feet  3  inches,  and 
depth  1  foot  10  inches? 

Ans.  44/fc.  8in.  Ssec. 

Note,  To  find  the  cubic  feet,  or  solid  content  of  bales,  cases,  boxes, 
&c.  multiply  the  length  by  the  breadth,  and  that  product  by  the  thick- 
ness. 

10.  A  merchant  imports  from  London  six  bales  of  the  fol- 
lowing dimensions,  viz. 

Length.  Breadth.  Depth. 

Ft.   in.  Ft.  in.  Ft.  in 

No.  1.         2     10  24  19 

2.  2     10  26  13 

3.  36  22  18 

4.  2     10  28  19 

5.  2     10  26  19 

6.  2     11  28  18 

What  are  the  solid  contents,  and  how  much  will  the  freight 
amount  to,  at  20  dollars  per  ton  of  40  feet  ? 

Ans.  lift.  lin.  and  freight  35  dollars  79  cts. 

To  find  a  ship's  tonnage  by  Carpenter's  measure. 

For  single  decked  vessels,  multiply  the  length,  breadth 
at  the  main  beam,  and  depth  of  the  hold  together,  and  di- 
vide the  product  by  95 ;  but  if  the  vessel  be  double  decked, 
take  half  the  breadth  of  the  main  beam  for  the  depth  of  the 
hold,  and  work  as  for  a  single  decked  vessel. 

O 


158  MULTIPLICATION  OF  DUODECIMALS 

. 

EXAMPLES. 

1.  The   length  of  a  single  decked  vessel  is  60'  feet,  the 
breadth  20,  and  depth  10;  what  is  the  tonnage? 

Then  60x20x10  =  12000 
And  1  2000  ~  95  =  1  26  fy  tons.    An*. 
Or,  as  95    :    20x10    ::    60    :    126r<V  Ans. 

2.  Required  the  tonnage  of  a  double  decked  vessel,  whose 
'ength  is  90,  and  breadth  30. 

Then  90  X  30  X  15  (half  breadth)  =40500. 

And  40500-^-95=426-,^  tons.  Ans. 
Or,  as  95  :  30  X  15  ::  90  :  426T\  Ans. 

3.  A  single  decked  vessel  is  64  feet  long,  22  feet  broad, 
and  10  feet  deep;  what  is  its  tonnage? 

Ans.  148T4y-  tons. 

4.  What  will  be  the  tonnage  of  a  double  decked  vessel 
whose  length  is  80  feet,  and  breadth  26  feet  ? 

Ans.  284}|  tons- 


To  find  the  Government  tonnage. 

"  If  the  vessel  be  double  decked,  take  the  length  thereof 
from  the  fore  part  of  the  main  stem,  to  the  after  part  of  the 
stern  post,  above  the  upper  deck  ;  the  breadth  thereof  at  the 
broadest  part  above  the  main  wales,  half  of  which  breadth 
shall  be  accounted  the  depth  of  such  vessel,  and  then  deduct 
from  the  length  three-fifths  of  the  breadth  ;  multiply  the 
remainder  by  the  breadth,  and  the  product  by  the  depth, 
and  divide  this  last  product  by  95,  for  the  tonnage.  But  if 
it  be  a  single  decked  vessel,  take  the  length  and  broadth, 
as  directed  above;  deduct  from  the  said  length  three-fifths 
of  the  breadth,  and  take  the  depth  from  under  the  side  of 
the  deck  plank  to  the  ceiling  in  the  hold,  then  multiply  and 
divide  as  aforesaid,  and  the  quotient  shall  be  deemed  the 
tonnage." 


CARPENTERS'  OR  SLIDING  RULE.  159 

SECTION  2. 

The  Carpenters'  or  Sliding  Rule. 

THIS  rule  is  not  only  useful  in  measuring  timber,  artifi- 
cers' work,  and  taking  dimensions,  but  also  in  ascertaining 
the  contents  of  such  work  ;  it  is  therefore  a  rule  which  all 
mechanics,  having  any  thing  to  do  with  mensuration,  ought 
to  possess  and  understand. 

It  consists  of  two  equal  pieces  of  box,  each  one  foot  in 
length,  connected  together  by  a  folding  joint ;  in  one  of  these 
equal  pieces,  there  is  a  slider,  and  four  lines  marked  at  the 
right-hand  with  the  letters  A,  B,  C,  D  ;  two  of  these  lines 
are  upon  the  slider,  and  the  other  two  upon  the  rule.  Three 
of  these  lines,  viz.  A,  B,  C,  are  called  double  lines,  because 
they  proceed  from  1  to  10  twice  over ;  these  three  lines  are 
alike  both  in  number  and  division.  They  are  numbered 
from  the  left-hand  towards  the  right  with  the  figures  1,  2,  3, 

5,  6,  7,  8,  9,  then  1,  which  stands  in  the  middle;  the 
numbers  then  proceed,  2,  3,  4,  5,  6,  7,  8,  9,  and  10,  which 
stands  at  the  right-hand  end  of  the  rule.  These  numbers 
have  no  determinate  value  of  their  own,  but  depend  upon 
the  value  you  set  upon  the  unit  at  the  left-hand  of  this  part 
of  the  rule ;  thus,  if  you  call  it  1,  the  1  in  the  middle  will 
be  10,  the  other  figures  which  follow,  will  be  20,  30,  &c. 
and  the  10  at  the  right-hand  will  be  100,  If  you  call  the 
first  or  left-hand  unit.  10,  the  middle  1  will  be  100,  and  the 
following  figures  will'  be  200,  300,  &c.  and  the  10  at  the 
right-hand  end  will  be  1000.  Or  if  you  call  the  first  or 
left-hand  unit  100,  the  middle  1  will  be  1000,  the  following 
figures  2000,  3000,  &c.  and  the  10  at  the  right-hand  10,000. 
Lastly,  as  you  alter,  or  number  the  large  divisions,  so  you 
must  alter  the  small  divisions  in  the  same  proportion. 

The  fourth  line  D  is  a  single  line,  proceeding  from  4  to 
40 ;  it  is  also  called  the  girt  line,  from  its  use  in  casting  up 
the  contents  of  timber.  Upon  it  are  marked  W  G  at  17,  15, 
and  A  G  at  18,  95,  the  wine  and  gauge  points,  to  make  it 
serve  the  purpose  of  a  gauging  rule. 

Tne  use  of  the  double  lines,  A  and  B,  is  for  working  the 
rule  of  proportion,  and  finding  the  areas  of  plane  figures. 
On  the  other  part  of  this  side  of  the  rule  there  is  a  table  to 


160  CARPENTERS'  OR  SLIDING  RULE. 

ascertain  the  value  of  a  ton,  or  50  cubic  feel  of  timber  at  all 
prices  from  6  pence  to  24  pence  per  foot. 

On  the  other  side  of  the  rule  are  several  plain  scales 
divided  into  12th  parts,  marked  mrA,  ^>  i»  ?,  &c.  signifying 
that  the  inch,  5  inch,  &c.  are  divided  into  1:2  parts.  These 
scales  are  useful  for  planning  dimensions,  that  are  taken  in 
feet  and  inches.  Again,  the  edge  of  the  rule  is  divided  into 
inches,  and  each  of  these  into  eight  parts,  representing  half 
inches,  quarter  inches,  and  half  quarters. 

In  this  description  the  rule  is  folded ;  but  when  it  is  opened 
and  the  slider  drawn  out,  the  hack  part  will  be  ibund  divided 
like  the  edge  of  the  rule,  so  that  all  together  will  measure 
3  feet  or  one  yard. 

USE  OF  THE  CARPENTERS'  RULE. 
1st.   To  multiply  numbers  together. 

EXAMPLE. 

1.  Suppose  the  two  numbers  13  and  24. 

Set  1  on  B  to  13  on  A ;  then  against  24  on  B,  stands  312 
on  A,  which  is  the  required  product  of  the  two  given  num- 
bers 13  anu  24. 

Note.  In  any  operation  when  a  number  runs  beyond  the  end  of  the 
line,  seek  it  on  the  other  radius,  or  other  part  of  the  line  ;  that  is,  take 
the  10th  part  of  it,  or  the  100th  part  of  it,  &c.  and  increase  the  product 
of  it  proportionally  10  fold,  or  100  fold,  <fec. 

2.  Multiply  12  by  16  Ans.  192. 

3.  35        19  665. 

4.  270        54  14580. 

2d.  Division  of  numbers  by  the  Carpenters'  Rule. 
EXAMPLE. 

1.  Required  to  divide  360  by  12. 

Set  the  divisor  12  on  B  to  the  dividend  360  on  A  ;  then 
against  1  on  B  stands  30,  the  quotient  on  A. 

2.  Divide  665  by  19.  Quotient     35. 

3.  396     "   27.  14,6. 

4.  741         42.  17,6. 

5.  76HO        24.  320. 
Note.     In  this  last  example,  because  7G80  is  not  contained  on  A,  one- 

lenth  of  the  number,  viz.  768  is  taken,  to  make  it  fall  within  the  com- 
pass of  the  scale — The  quotient  of  this  sum  is  32,  but  as  the  dividond 
w;is  diminished  by  a  division  of  1(1,  so  the  «juotieiit  nmst  be  multiplied 
uy  the  Ran»e  number,  arid  32  10—320. 


CARPENTERS'  OR  SLIDING  RULE.  161 

3d.   To  square  numbers  by  the  Carpenters'  Rule. 
EXAMPLE. 

1.  Required  to  square  the  number  25. 

Set  1  or  100  on  B  to  the  10  on  D  ;  then  against  every 

number  on  D  stands  its  square  on  the  line  C.  Thus  against 
25  on  D,  stands  625,  its  square  on  C. 

2.  Required  the  square  of  30.  Ans.  900. 

3.  Required  the  square  of  35.  1225. 

4.  Required  the  square  of  40.  1600. 

Note.  If  the  given  number  be  hundreds,  &c.  reckon  the  1  on  D  for 
100,  or  1000,  &c.  then  the  corresponding  one  on  C  is  10000,  or  100000, 
&c. ,  thus  the  square  of  230  is  found  to  be  52900. 

4th.   To  find  a  fourth  proportional  to  three  numbers :  or  to 
perform  the  Rule  of  Three  by  the  Carpenters'  Rule. 

EXAMPLE. 

1.  Required  to  find  a  fourth  proportional  to  12,  28,  and 
114.  * 

Set  the  first  term  12  on  B  to  the  second  term  28  on  A ; 
then  against  the  third  term  114  on  B,  stands  266  on  A, 
which  is  the  fourth  proportional  sought. 

2.  Required  the  fourth  proportional  term  to  the  numbers 
25  :  75  :  :  100.  Ans.   300. 

3.  Required  the  fourth  proportional  term  to  the  numbers 
27  :  20  : :  73.  Ans.  54¥2T 

5th.   To  extract  the  Square  Root  of  any  number  by  the 
Carpenters1  Rule. 

EXAMPLE. 

1.  Required  the  square  root  of  400, 

Set  1  upon  C  to  10  upon  D  ;  then  against  the  number 
400  on  C,  stands  its  root  20  on  D. 

2.  Required  the  square  root  of  529.  Ans.  23. 

3.  What  is  the  square  root  of  900  ?  30. 

4.  What  is  the  square  root  of  300  ?  17,3 -f 

O  2 


162  MEASURfNG  OF  BOARDS  AND  TIMBKR. 

SECTION  3. 

Measuring  of  Boards  and  Timber. 

1st     To  find  ike  superficial  content  of  a  board  or  platik. 

RULE. 

Multiply  the  length  by  the  mean  breadth.  When  the 
board  is  broader  at  one  end  than  the  other,  add  the  brewlih 
of  the  two  ends  together,  and  take  half  the  sum  for  tin..- 
mean  breadth. 

By  the  Carpenters'  Rule. 

Set  12  on  B  to  the  breadth  in  inches  on  A;  then  against 
the  length  in  feet  on  B,  you  will  find  the  superficies  on  A, 
in  feet. 

EXAMPLES. 

1.  How  many  feet  are  there  in  a  board  that  is  13  {(-el 
long  and  16  inches  broad  ? 

Operation. 

By  duodecimals.  By  decimals. 

'Ft.  in.  13, 

13     0  1,33 

1      4  


39 

13     0  39 

440  13 


17     4     0     Ans.  17,29     Ans. 

By  the  Carpenters'  Rule — 
As  12  on  B  :  16  on  A  : :  13  on  B  :  17£  on  A.     Ans. 

2.  Required   the   superficies   of  a   board,   whose    mean 
breadth  is  1  foot  2  inches,  and  length  12  feet  6  inches? 

Ans.  14  feet  7  inches. 

3.  Required  the  value  of  5  oaken  planks,  at  3  cents  pei 
foot ;  each  of  them  being  17£  feet  long,  and  their  several 
breadths  as  follows,  viz.  ;  two  of  13£  inches  in  the  middle, 
one  of  14£   inches  in  the  middle,  and  the  two  remaining 
ones,  car-h  18  inches  at  the  broader  end,  and  11:1  inHvs  at 
the  narrower  1  Ans.   3  dolls.  9i  cts. 


MEASURING  OF  BOARDS  AND  TIMBER.  163 

2d.  Having  the  breadth  of  a  board  or  plank  in  inches, 
tojind  how  much  in  length  will  make  afoot^  or  any  other 
assigned  quantity. 

RULE. 

Divide  144,  or  the  area  to  be  cut  off,  by%ie  breadth'  in 
inches,  and  the  quotient  will  be  the  length  in  inches. 

EXAMPLE. 

1.  How  many  inches  in  length  will  it  require  to  make  one 
foot,  of  a  board  that  is  9  inches  broad? 

Operation.    144-1-9=16  inches,  the  length  required. 

2.  How  many  inches  in  length,  of  a  board   that  is  23 
inches  wide,  will  make  1  foot? 

Ans.  6,26 -f  inches. 

3.  From  a  mahogany  plank  26  inches  broad,  a  yard  and 
a  half  (or  13  feet  6  inches)  is  required  to  be  cut  off*;  what 
distance  from  the  end  must  the  line  be  struck  ? 

Ans.  74,7692  inches,  or  6,23  feet. 

3d.     To  find  the  solid  content  of  squared  or  four  sided 
timber. 

RULE. 

Multiply  the  mean  breadth  by  the  mean  thickness,  and 
that  product  by  the  length ;  the  last  product  will  give  the 
solid  content. 

Note.  1.  If  the  tree  taper  regularly  from  one  end  to  the  other,  take 
the  mean  breadth  and  thickness  in  the  middle ;  or  take  half  the  sum 
of  the  dimensions  at  the  two  ends,  for  the  mean  dimension. 

2.  If  the  piece  does  riot  taper  regularly,  take  several  different  di- 
mensions, add  them  all  together,  and  divide  their  sum  by  their  num- 
ber, for  the  mean  dimension. 

3.  The  quarter  girt  is  an  arithmetical  mean,  proportional  between 
the  mean  breadth  and  thickness,  that  is  the  square  root  of  the  pro- 
duct. 

EXAMPLES- 

1.  If  a  piece  of  timber  be  2  feet  9  inches  deep,  and  1  foot 
7  inches  broad,  and  the  length  16  feet  9  inches,,  (or  which 


164  MEASURING  OF  BOARDS  AND  TIMBER. 

is  the  same  thing,*  if  the  quarter  girt  be  26  inches,  and  the 
length  16  feet  9  inches,)  how  many  solid  feet  are  contained 
therein  ? 

Operation. 

26  inches  quarter  girt  16,75  — 16  feet  9  inches,  the  length. 

26  676 


156  10050 

52  11725 

10050 

676  square  

144)  11323,00(78,63  + feet.    Arts. 
1008 


1243 
1152 

910 
864 

460 
432 

28  rem. 

By  the  Carpenter's  Rule — 
As  12  on  D  :  16f  on  C  : :  26  on  D  :  78£  on  C.  Ans. 

2.  The  quarter  girt  of  a  piece  of  squared  timber  is  15 
inches,  and  the  length  18  feet;  required  the  solidity? 

Ans.  28  J  feet. 

3.  If  a  piece  of  squared  timber  be  25  inches  square  at 
the  greater  end,  and  9  inches  square  at  the  less,  and  the 
length  be  20  feet ;  what  is  the  solid  content  ? 

Ans.  40,13  feet. 

4.  Suppose  a  piece  of  squared  timber  to  measure  32  by 
20  inches  at  the  greater  end,  and  10  by  6  inches  ai  the  less, 
and  the  length  18  feet;  how  many  feet  of  timber  are  con- 
tained therein?  Ans.  34,12 -f  feet. 

*  This  is  making-  use  of  an  arithmetical  mean,  instead  of  a  geomet- 
rical one,  which  is  not  exactly  true;  it  is,  however,  sufficiently  exact 
for  common  purposes,  when  the  timber  is  nearly  square;  the  error  in- 
creases, the  more  the  breadth  and  depth  differ  from  each  other. — 
When  greater  exact  ness  is  required,  multiply  the  breadth  by  the 
depth  in  the  middle,  and  that,  product  by  the  length,  ibr  the  true  con- 
tent. 


MEASURING  OF  BOARDS  AND  TIMBER.  165 

4:th.    To  find  the  solid  content  of  round  timber. 
RULE. 

Multiply  the  square  of  the  quarter  girt,  or  of  ^  of  the 
mean  circumference,  by  the  length,  for  the  content. 

Note.  1.  To  find  the  quarter  girt  of  round  timber,  measure  round 
the  middle  with  a  line,  one-fourth  part  of  this  is  reckoned  the  quarter 
girt 

2.  When  the  tree  is  tapering1,  take  either  the  mean  dimension,  as  in 
squared  timber,  or  girt  it  at  both  ends,  and  take  half  the  sum.     If  the 
tree  is  very  irregular,  divide  it  into  several  lengths,  add  all  the  girts  to- 
gether, and  divide  the  amount  by  the  sum  of  them,  for  tho  mean  girt. 

3.  The  buyer  is  allowed  to  take  the  girt  anywhere  between   the 
greater  end  and  the  middle,  if  it  taper ;  an   allowance  must  also  be 
made  for  bark  ;  one-tenth  for  oak,  but  less  for  ash,  beech,  &c. 

EXAMPLES. 

1.  A  piece  of  round  timber  being  9  feet  6  inches  long,, 
and  its  mean  quarter  girt  42  inches,  what  is  the  content? 

Operation. 

Decimals.  Duodecimals. 

3,5—42  inches,  quarter  girt.          Ft.  in. 

3,5  3     6=42  inches. 
3     6 


175  

105  190 
10     6 


12,25 


9,5  length  12     3     0 

9     6 


6125  

11025  6     1 
110     3 


116,375  content. 

116     4     6 

By  the  Carpenters'  Rule — 

As  9,5  on  C  :  10  on  D  : :  3,5  on  D  :  116J  on  C  (  4 
Or,9,5  on  C  :  12  on  D  : :  42   on  D  :  116i  on  C  $ 
2.  The  length  of  a  tree  is  24  feet,  its  girt  at  the  thicker 
end  is  14  feet,  and  at  the  smaller  end  2  feet;  what  is  its 
content?  Ans.  96  fert. 


166  CARPENTERS'  AND  JOINERS'  WORK- 

3.  If  a  piece  of  round  timber  18  feet  long,  measure  00 
inches  in  circumference,  or  the  quarter  girt  24  inches ;  how 
many  feet  of  timber  does  it  contain? 

Ans.  72  feet. 

4.  If  a  piece  of  round  timber  measure  11  feet  4  inches 
at  the  larger  end,  2  feet  8  inches  at  the  less,  and  its  length 
21  feet,  how  many  feet  of  timber  are  contained  therein? 

Ans.  64,31  feet. 

5.  Required  the  amount  of  three  pieces  of  round  timb<  r 
measuring  as  follows,  viz. 

The  first  24     feet  long  and  mean  girt     8  feet, 

The  second      14*         do.  do.  3,15 

The  third          17i         do.  do.          6,28 

Ans   147  ft  4.  feet. 


SECTION  4. 
OF  CARPENTERS'   AND  JOINERS'  WORK. 

To  this  branch  belongs  all  the  wood-work  of  a  house, 
such  as  framing,  flooring,  partitioning,  roofing,  &c. 

Carpenters  usually  measure  their  work  by  the  square, 
(consisting  of  100  superficial  feet)  the  yard  or  foot;  but 
enriched  mouldings,  cornices,  &c.  are  estimated  by  run- 
ning or  lineal  measure,  and  some  things  are  rated  by  the 
piece. 

In  measuring  of  Carpenter's  work,  the  string  is  made  to 
ply  close  to  every  part  of  the  work  over  which  it  passes. 

Partitions  are  measured  from  wall  to  wall  for  one  dimen- 
sion, and  from  floor  to  floor,  as  far  as  they  extend,  for  the 
other. 

In  framing,  no  deductions  are  made  for  door-ways,  fire- 
piacrs,  or  other  vacancies,  on  account  of  the  additional 
trouble  of  framing  arising  from  them. 

For  stair-case's,  take  the  breadth  of  all  the  steps,  by 
milking  a  line  ply  close  over  them,  from  the  top  to  the  bot- 
inm,  and  multiply  the  length  of  this  line,  by  the  length  of 
thr  step,  for  the  whole  axea.  T>y  Iho  length  of  a  step  is 


CARPENTERS'  AND  JOINERS'  WORK.  167 

meant  the  length  of  the  front,  and  the  returns  at  the  two 
ends  ;  and  by  the  breadth,  is  to  be  understood  the  girt  of  its 
two  outer  surfaces,  or  the  tread  and  rise. 

The  rail  of  a  stair-case  is  taken  at  so  much  per  foot  in 
length,  according  to  the  diameter  of  the  well-hole ;  archi- 
trave string  boards,  by  the  foot  superficial ;  brackets  and 
strings  at  so  much  per  piece,  according  to  the  workmanship. 

Wainscoting  is  measured  by  the  yard  square,  consisting 
of  9  feet. 

Door  cases,  frame  doors,  modillion  cornices,  eaves,  frontis- 
pieces, <fec.  are  generally  measured  by  the  foot  superficial. 

Joists  are  measured  by  multiplying  their  breadth  by  their 
depth,  and  that  product  by  their  length.  They  receive  va- 
rious names,  according  to  the  place  in  which  they  are  laid 
to  form  a  floor  ;  such  as  trimming  joists,  girders,  binding 
joists,  bridging  joists,  ceiling  joists,  &c. 

In  boarded  flooring,  the  dimensions  must  be  taken  to  the 
extreme  parts  ;  and  the  number  of  squares  of  100  feet,  are 
to  be  .  calculated  from  these  dimensions.  Deductions  must 
be  made  for  chimneys,  stair-cases,  &c. 

In  roofing,  take  the  whole  length  of  the  timber,  for  the 
length  of  the  framing,  and  gird  over  the  ridge  from  wall  to 
wall  with  a  line,  for  the  breadth.  This  length  and  breadth 
multiplied  together  give  the  content. 

In  measuring  of  roofing  for  work  and  materials,  all  holes 
for  chimney-shafts,  sky-lights,  &c.  are  included  in  the  mea- 
surement, on  account  of  their  trouble  and  waste  of  materials  ; 
but  for  workmanship  alone,  they  are  generally  deducted. 

It  is  a  common  rule  among  carpenters,  that  the  flat  of 
any  hou.se,  and  half  the  flat  thereof  taken  within  the  walls, 
ks  equal  to  the  measure  of  the  roof  of  the  same  house  ;  this 
is,  however,  only  when  the  roof  is  the  true  pitch — where 
the  length  of  the  rafters  are  5  of  the  breadth  of  the  build- 
ing. The  pitch  of  roofs  varies  according  to  the  materials 
with  whio.h  they  are  covered,  and  fancy  of  the  builder. 

Weather- boarding,  like  flooring,  is  measured  by  the 
square,  arid  sometimes  by  the  yard. 


168  CARPENTERS'  AND  JOINERS'  WORK. 

EXAMPLES. 

1.  If  a  floor  be  57  feet  3  inches  long,  and  28  feet  6  inches 
broad,  how  many  squares  of  flooring  does  it  contain  ? 

Operation. 

By  decimals.  By  duodecimals. 
Ft.   in.  Ft.     in. 

57     3=57,25  57       3 

28     6=   28,5  28       6 


28625  456 

45800  114 

11450  28  7  6 


100)1631,625 
16,31,625 


700 


100)1631     7     6 

16,31    7     6 
Ans.  16  squares  31  feet  7  in.  6' 

2.  Let  a  floor  be  53  feet  6   inches  long,  and  47  feet  9 
inches  broad,  how  many  squares  does  it  contain  ? 

Ans.  25  squares  54  feet. 

3.  A  floor  being  36  feet  3  inches  long,  and  16  feet  6 
inches  broad,  what  will  it  cost  at  4  dollars  and  50  cents  per 
square?  Ans.  26  dolls.  91  cents. 

4.  A  room  is  35  feet  long  and  30  feet  wide ;  there  is  in  it 
a  fire-place  which  measures  6  feet  by  4  feet  6  inches,  and  a 
well-hole  for  the  stairs  measures  10  leet  6  inches  by  8  feet  ; 
what  will  the  flooring  come  to  at  3  dollars  and  75  cents  per 
square?  Ans.  35  dolls.  21 -f  cts,. 

5.  How  many  squares  are  contained  in  a  partition  that  i* 
82  feet  6  inches  long,  and  12  feet  3  inches  high? 

Ans.  10  squares  and  10-ffeet. 

6.  If  a  partition  between  rooms  be  in  length  91   feet  9 
inches,  and  its  height  11  feet  3  inches;   how  many  squares 
are  contained  in  it,  and  how  much  does  it  come  to  at  4  dol 
lars  and  50  cents  per  square  ?  * 

Ans.  10  squares  32  feet,  and  costs  46  dolls.  44  cts. 


CARPENTERS    AND  JOINERS    WORK-  169 

7.  If  a  house  within  the  walls  be  44  feet  6  inches  long, 
and  18  feet  3  inches  broad  ;  how  many  squares  of  roofing 
will  it  contain,  allowing  the  roof  to  be  the  true  pitch  ? 

Operation. 

By  decimals.  By  duodecimals. 

Ft.         Ft.  in.  18     3 

18,25=18     3  the  breadth.       44     6 
44,5=44     6  the  length. 


72 

9125                 72 
7300  


7300  792 
11  0  0" 


Flat     812,125  9     1     6 

Half    406,062 


Flat     812     1 
-100)1218,187  Half    406  + 


Sum  12,18+  +  100)1218     1     6 


Sum  12,18 

Ans.  12  sq.  18  ft. 

8.  What  cost  the  roofing  of  a  house  at  1  dollar  and  40 
cents  per  square  ;  the  length  within  the  walls  being  52  feet 
8  inches,  and  the  breadth  3-0  feet  6  inches  ;  the  roof  being 
of  a  true  pitch  ? 

Ans.  33  dollars  73  cents. 

9.  Suppose  a  house  measures,  within  the  walls,  40  feet  6 
inches  in  length,  and  20  feet  6  inches  in  breadth,  and  the 
roof  being  a  true  pitch  ;  how  many  squares  of  roofing  does 
it  contain,  and  how  much  will  it  cost  at  2  dollars  25  cents 
per  square  ? 

Ans.  12,45375  squares,  and  costs  28  dolls.  2  +  cts. 

Note.  All  timbers  in  a  roof  are  measured  in  the  same  manner  as 
in  floors,  except  king-posts,  which  are  measured  by  taking  their 
breadth  and  depth  at  the  widest  place,  arid  multiplying  these  together., 
and  the  product  by  the  length. 

10.  If  a   room  or  wainscot,  being  girt  downwards  over 


!?(>  BIUCKLAYEKS'  WORK.     * 

the  mouldings,  be   15   feet  9  inches  high,  and   126   feet  3 
in* -lies  in  compass  ;  how.  many  yards  does  that  room  contain  '/ 

Operation. 
By  decimals*  By  duodecimals. 

12V5  ' Ft-  in- 

15,75  Ig    3 


63125  63Q 

88375  126 

63125 


12655  189° 

63    1     6 

31     6    9 

9)1988,4375  390 


Sum  220,8 

Sum  220     8 

Ans.  220  yds.  8  feet. 

11.  If  a  room  of  wainscot  be  16  feet  3  inches  high,  and 
the  compass  of  the   room   137  feet  6  inches  ;  how  many 
yards  are  contained  in  it  ?  Ans.  248  yards  2  -f  feet. 

12.  If  the  window-shutters  about  a  room  be  69  feet  9 
inches   broad,  and  6  feet  3  inches  high  ;  how  many  yards 
are  contained  therein,  at  work  and  half? 

Ans.  72,656  yards. 

13.  What  will  the  wainscoting  of  a  room  come  to  at  80 
cents  per  square  yard,  supposing  the  height  of  the  room,  in- 
cluding the  cornice  and  moulding,  be  12  feet  6  inches,  and 
the  compass  83  feet  8  inches  ;  three  window-shutters,  each 
7  feet  8  inches  by  two  feet  6  inches,  and  the  door  7  feet  by 
3  feet  6  inches  ;  the  shutters  and  door  being  worked  on  both 
sides,  are  reckoned  work  and  half? 

Ans.  96  dollars  60J  cents. 


SECTION  5. 

OF  BRICKLAYERS'  WORK- 

BKICK  WOKK  is  measure.  I  ;ind  rsfimatr'd  in  various  ways. 
!i  somn  places  walls  aiv  mrasim-d  !>y  fho  rod  square  of 
1  i  frc'i  ;  so  th.-.t  on'-  rod  in  length,  and  one  in  breadth 


BRICKLAYERS'  WORK.  171 

contain  272,25  square  feet ;  in  other  places  the  custom  is  to 
allow  18  feet  to  the  rod,  that  is,  324  square  feet. 

In  other  places  they  measure  by  the  rod  of  21  feet  long, 
and  3  feet  high,  that  is,  62  square  feet.  Again,  in  other 
places  they  account  16^  feet  long  and  1  foot  high,  that  is, 
16^  square  feet,  a  rod  or  perch;  or  again,  by  the  yard  of 
9  square  feet ;  and  oftentimes  the  work  is  estimated  at  so 
much  per  thousand  bricks. 

When  brick  work  is  measured  by  the  rod,  or  perch,  it 
must  be  estimated  at  the  rate  of  a  brick  and  a  half  thick; 
so  that  if  a  wall  be  more  or  less  than  this  standard  thickness, 
it  must  be  reduced  to  it  by  the  following 

RULE. 

Multiply  the  superficial  content  of  the  wall  by  the  num- 
ber of  half  bricks  in  the  thickness,  and  divide  the  product 
by  3  for  the  superficial  feet  in  standard  thickness. 

EXAMPLES. 

If  a  wall  be  72  feet  6  inches  long,  and  19  feet  3  inches 
high,  and  5  bricks  and  a  half  thick ;  how  many  rods  of 
brick  work  are  contained  therein,  when  reduced  to  the 
standard  thickness  ? 

Operation. 

By  decimals.  By  duodecimals. 

19,25  =  the  height  Ft.    in. 

72,5  =  the  length.  72     6 

"9625  19     3 

3850  648 

13475      -  72 

139^625  1368"" 

11=  the  thickness.  18     1     6' 

-f-3)15351,875  9     6 

—272,25)  5117,291  (18  rods.  1395     7  ^6 

239479  i- 

68,06)  216/79(  3  quarters.  -3)15351     10     6 

lalT  272)511^  (18  rote. 

2397 
68)  "22f  (  3  quarters 

Tffeet. 

Note.  Observe  that  68,06  is  the  one- fourth  part  of  272,25,  and  68  is 
only  the  one-fourth  part  of  272.  As  the  number  272^  is  an  inconve- 
nient number  to  divide* by,  the  \  is  usually  ornitfed,  and  the  content  in 
feet  divided  only  by  272 ;  the  difference  being  too  trifling  to  be  con- 
sidercd  in  practice. 


172 


BRICKLAYERS'  WORK. 


To  find  fixed  divisors  for  bringing  the  answer  into  feet  or 
rods  of  a  standard  thickness,  without  multiplying  the  su- 
perficies by  the  number  of  half  bricks,  <$*c. 

RULE. 

Divide  three,  the  number  of  half  bricks  in  l£,  by  the 
number  of  half  bricks  in  the  thickness,  the  quotient  will  be 
a  divisor,  which  will  give  the  answer  in  feet.  Or  if  a  divi- 
sor is  sought  for,  that  will  bring  the  answer  in  rods  at  once, 
multiply  272  by  the  divisor  found  for  feet,  and  the  product 
will  be  a  divisor  for  rods ;  as  in  the  following 

TABLE. 


1 

The  thickness  of 
the  wall. 

2 

Divisors  for  the 
ansiver  in  feet. 

3 

Divisors  for  the 
ansiver  in  rods. 

1     brick 

1, 

408 

l£  brick 

1 

472 

2     bricks 

,75 

204 

2J  bricks 

,6 

163,2 

3     bricks 

,5 

136 

3i  bricks 

,4285 

116,6 

4     bricks 

,375 

102 

4^  bricks 

,3333 

90,6 

5     bricks 

,3 

81,6 

5J   bricks 

,2727 

74,18 

Application  of  the  above  Table. 

Multiply  the  length  of  the  given  wall  bv  the  breadth ,  o 
serve  the  number  of  half  bricks  it  is  in  thickness ;  and  op- 
[>rsite  thereto  will  be  found  in  the  second  column  the  di- 
visor to  reduce  it  to  feet  ana  in  the  third  column  the  divi- 
sor for  rods.  Thus  in  the  above  example  72,5  X  by  19,25  — 
1395,625. 

Ana   1395,620  — 2727  =  5117+    the    number   of  feet   in 
standard  nn-asure. 

And  i:W5,(>-jr>-74,18^18,8-f  th<>  numl>cr  of  rods 


BRICKLAYERS'  WORK.  173 

Or,  by  the  Carpenters'  Rule — 

As  the  tabular  divisor,  against  the  thickness  of  the  wall 
:  is  to  the  length  of  the  wall  : :  so  is  the  breadth  :  to  the 
content. 

As  74,18  on  B  :  72,5  on  A  ::  19,25  on  B  :  18  j  on  A.  Ans. 

To  find  the  dimensions  of  a  building,  measure  half  around 
on  the  outside,  and  half  on  the  inside,  for  the  whole  length 
of  the  wall ;  this  length  being  multiplied  by  the  height  gives 
the  superficies.  All  the  vacuities,  such  as  doors,  windows, 
window -backs,  &c.  must  be  deducted,  for  materials ;  but  for 
workmanship  alone  no  deductions  are  to  be  made,  and  the 
measurement  is  usually  taken  altogether  on  the  outside. 
This  is  done  in  consideration  of  the  trouble  of  the  returns 
or  angles.  There  are  also  some  other  allowances,  such  as 
double  measure  for  feathered  gable  ends,  &c. 

2.  How  many  yards  and  rods  of  standard  thickness  are 
contained  in  a  brick  wall,  wWbse  length  is  57  feet  3  inches, 
and  height  24  feet  6  inches ;  the  wall  being  2J  bricks  thick  ? 

Ans.  259,74  yards,  or  8,58  +  rods. 

3.  If  a  wall  be  245  feet  9  inches  long,   16  feet  6  inches 
high,  and  2|  bricks  thick;  how  many  rods  "of  brick  work 
are  contained  therein,  when  reduced  to  standard  thickness  ? 

Ans.  24  rods  3  quarters  24  feet. 

4.  A  triangle  gable  end  is  raised  to  the  hei^     of  15  feet 
above  the  wall  of  a  house,  whose  width  is  45  feet  and  the 
thickness  of  the  wall  2£  bricks  ;  required  the  contei^  in  rods 
at  standard  thickness?  Ans.  2  rods  18  feet. 

Chimneys  by  some  are  measured  as  if  they  were  solid,  de- 
ducting only  the  vacuity  from  the  hearth  to  the  mantle,  on 
account  of  their  trouble. 

But  by  others,  they  are  girt  or  measured  round  for  their 
breadth,  and  the  height  of  their  story,  taking  the  depth  of 
the  jambs  for  their  thickness.  And  in  this  case  no  deduc- 
tion is  made  for  the  vacuity  from  the  floor  to  the  mantle-tree, 
because  of  the  gathering  of  the  breast  and  wings,  to  make 
-oom  for  the  hearth  in  the  next  story. 
P2 


174  MASONS'  WORK. 

If  the  chimney  back  be  a  party  wall,  and  the  wall  be 
measured  by  itself,  then  the  depth  of  the  two  jambs  and 
length  of  the  breast  is  to  be  taken  for  the  length,  and  the 
height  of  the  story  for  the  breadth,  at  the  same  thickness 
with  the  jambs. 

Those  parts  of  the  chimney-shaft  which  appear  above 
the  roof  are  to  be  girt  with  a  line  round  about  the  least  place 
of  them  for  the  length,  and  take  the  height  for  the  breadth  ; 
and  if  they  are  4  inches  thick,  they  are  to  be  accounted  as 
one  brick  work,  and  if  they  are  9  inches  thick,  they  are  to 
be  taken  for  l£  brick  work,  on  account  of  the  trouble  of 
plastering  and  scaffolding. 

It  is  customary  in  most  places  to  allow  double  measure 
for  chimneys. 


SECTION  6. 


OF  MASONS'  WORK. 

MASONS'  work  is  measured  sometimes  by  the  foot  solid, 
sometimes  by  the  foot  superficial,  and  sometimes  by  the  foot 
in  length.  It  is  also  measured  by  the  yard,  and  mostly  by 
the  rod  or  perch,  which  is  16£  feet  in  length,  18  inches  in 
breadth,  and  12  inches  in  depth. 

Walls  are  measured  by  the  perch;  columns,  blocks  of 
stone,  or  marble,  &c.  by  the  cubic  foot;  and  pavements, 
slabs,  chimney-pieces,  &c.  by  the  superficial  or  square  foot. 

Cubic,  or  solid  measure,  is  always  used  for  materials,  but 
square  measure  generally  for  workmanship. 

In  solid  measure,  the  true  length,  breadth  and  thickness 
are  taken  and  multiplied  into  each  other  for  the  content. 

!n  superficial  measure,  the  length  and  breadth  of  every 
part  of  the  projection,  which  is  seen  without  the  general 
upright  face  of  the  building,  is  taken  for  the  content. 


MASONS'  WOHK.  175 


EXAMPLES. 

1.  If  a  wall  be  97  leet  5  inches  long,  18  feet  3  inches 
high,  and  2  feet  3  inches  thick,  how  many  solid  feel,  and 
perches,  are  contained  therein  ? 

Operation. 

By  decimals.  By  uuodecimals. 
97,417  length  Ft.   in. 
18,25  breadth  97     5 
18     3 


4870S5 


194834  776 
779336  97 
97417  24     4     3" 
600 


1777,86025  superficies  1     60 

2,25  thickness 


1777   10  3 

888930125  2 
355572050 

355572050                                              3555     8  6 
444     5  6 


4000,1855625  solidity. 


in  cubic  ft.  4000     2094  u. 
4000^-24,75^161,616-f  feet.    Ans. 

2  How  many  solid  feet  and  perches  are  contained  in  a 
wall  53  feet  6  inches  long,  12  feet  3  inches  high,  and  2  feet 
thick?  Ans.  1310,75  feet,  and  52,9595  rods. 

3.  If  a  wail  be  107  feet  9  inches  long,  and  20  feet  6  inches 
high,  how  many  superficial  feet  are  contained  therein  ? 

Ans.  2208  feet  10  inches. 

4.  If  a  wall  be  112  feet   3  inches  long,  and    16   feet   6 
inches   high,  how  many  superficial   rods,  each   63   square 
feet,  are  contained  therein  ?  Ans.  29  rods  25  feet. 

5.  What  is  a  marble  slab  worth,  whose  length  is  5  feet  7 
inches,  and  breadth   1   foot  10  inches,  at  80  cents  per  fool 
superficial  1  Ans.  8  dolls.  19  els* 


176  PLASTERERS    WOixK 

SECTION  7. 

OF  PLASTERERS'  WORK. 

PLASTERERS'  WORK  is  principally  of  two  kinds,  viz. — 
first,  plastering  upon  laths,  called  ceiling  ;  and  second,  plas- 
tering upon  walls,  or  partitions  made  of  framed  timber, 
called  rendering,  which  are  measured  separately. 

Plasterers'  work  is  usually  measured  by  the  yard  square, 
consisting  of  9  square  feet ;  sometimes  it  is  measured  by 
the  square  foot,  and  sometimes  by  the  square  of  100  feet. 

Enriched  mouldings,  cornices,  &c.  are  rated  by  running, 
or  lineal  measure.  In  arches,  the  girt  round  them  multiplied 
by  the  length,  is  taken  for  the  superficies. 

Deductions  are  to  be  made  for  doors,  chimneys,  windows, 
and  other  large  vacuities.  But  when  the  windows,  or  other 
openings,  are  small,  they  are  seldom  deducted,  as  the  plas- 
tered returns  at  the  top  and  sides  are  allowed  to  compensate 
for  the  vacuity. 

Whitewashing  and  coloring  are  measured  in  the  same 
manner  as  plastering. 

EXAMPLES. 

1.  If  a  ceiling  be  59  feet  9  inches  long,  and  24  feet  6 
inches  broad,  how  many  superficial  yards  of  9  square  feet 
does  it  contain  ? 

Operation. 

By  decimals.  By  dtfodecimals. 

Ft.  in.  Ft.   in. 

59     9=59,75  feet  59     9 

24     6=   24,5  do.  24     6 

29875  236 
23900  118 
11950  29  10     6" 
18     0     0 


-7-9)1463,875  feet 
Ans.  1 62,65 -f  yards 


-r  9)  1463   10      6 


162     5   10     6    An*. 


\VKHJS    WORK  177 

2  If  the  plastered  partitions  between  rooms  be  141  feet  6 
inches  about,  and  1 1  feet  3  inches  high,  how  many  yards 
(Jo  tiiey  contain?  Ans.  176,87  yards. 

3.  What  will  the  plastering  of  a  ceiling  come  to  at  15 
cents  per  yard,  allowing  it  to  be  22  feet  7  inches  long,  and 
13  feet  11  inches  Lroad  ?  Ans.  5  dolls.  20  cts. 

4;  The  length  of  a  room  being  20  feet,  its  breadth  14 
feet  6  inches,  and  height  1 0  feet  4  inches ;  how  many  yards 
of  plastering  does  it  contain,  deducting  a  fire-place  of  4  feet 
by  4  feet  4  inches,  and  two  windows,  each  6  feet  by  3  feet 
2  inches  ?  Ans.  73^T  yards. 

5.  The  length  of  a  room  is  14  feet  5  inches,  breadth  13 
feet  2  inches,  and  height  9  feet  3  inches,  to  the  under  side 
of  the  cornice,  which  projects  5  inches  from  the  wall,  on  the 
upper  part  next  the  ceiling  ;  required  the  quantity  of  render- 
ing and  plastering  ;  there  being  no  deductions  but  for  one 
door,  the  size  whereof  is  7  by  4  feet  ? 

Ans.  53  yds.  5  ft.  of  rendering,  and  18  yds.  5  ft.  ceiling. 

6.  The  circular  vaulted  roof  of  a  church  measures  105 
feet  6  inches  in  the  arch,  and  275  feet  5  inches  in  length  ; 
what  will  the  plastering  come  to  at  12  cents  per  yard  ? 

Ans.  387  dolls.  42  cts. 

7.  What  will  the  whitewashing  of  a  room  come  to  at  2 
cents  per  yard,  allowing  it  to  be  30  feet  6  inches  long,  24 
feet  9  inches  broad,  and  10  feet  high  ;  no  deductions  being 
made  for  vacuities?  Ans.  4  dolls.  13£  cts. 


SECTION  8. 

OF  PAVERS'  WORK. 

PAVERS  WORK  is  measured  by  the  square  yard,  consist- 
ing of  9  square  feet.  The  superficies  is  found  by  multiply- 
ing the  length  by  the  I  readth. 


178  FA1JNTKRS'  WORK. 

EXAMPLES. 

1.  What  cost  the  paving  of  a  street  225  feet  6  inches  long. 
and  60  feet  6  inches  wide,  at  30  cents  per  square  yard  ? 

By  decimals.  By  duodecimals. 

Ft.    in.  ^    in. 

225  6=225,5  feet 
60  6=   60,5  do. 


13500 

11275  US 

13530 


9)13642    9 


1515     7 
30 


-7-9  )  13642,75  superficial  ft. 

1515,86  yards. 

30 

26  =the  price  of  7  ft.  9  in. 

Ans.  454,7580  ~454jeT 

Ans.  454  dolls.  76  cts. 

2.  What  will  the  paving  of  a  foot-path  come  to  at  28 
cents  per  yard,  the  length  being  35  feet  4  inches,  and  the 
breadth  8  feet  3  inches  ?  Ans.  9  dolls.  33  cts. 

3.  What  cost  the  paving  of  a  court-yard  at  38  cents  per 
yard,  the  length  being  27  feet  10  inches,  and  the  breadth  14 
feet  9  inches  ?  Ans.  ?7  dolls.  33£  cts. 

4.  What  will   be   the   expense  of  paving  a   rectangular 
yard,  whose  length  is  63  feet,  and  breadth  45  feet,  in  which 
there  is  laid  a  foot-path  5  feet  3  inches  broad,  running  the 
whole  length,  with  broad  stones,  at  36  cents  a  yard ;  the 
rest  being  paved  with  pebbles,  at  30  cents  a  yard  ? 

Ans.  96  dolls.  70£  cts. 


SECTION  9. 

OF  PAINTERS'  WORK. 

PAINTERS'  WORK  is  computed  in  square  yards  of  9  feet 
Kvery  part  is  measured  where  the  color  lies  ;  and  the  mea 
suring  line  is  pressed  close  into  all  the  mouldings,  corners, 
&ic.  over  which  it  passes. 


PAINTERS'  WORK.  179 

Windows,  casements,  &c.  are  estimated  at  so  much  a 
piece ;  and  it  is  usual  to  allow  double  measure  for  carved 
mouldings,  &c. 

The  value  of  painting  is  rated  by  the  number  of  coats ; 
or  whether  once,  twice,  or  thrice  colored  over,  and  the  dif- 
ferent qualities  and  costliness  of  the  colors. 

EXAMPLES. 

1,  How  many  yards  of  painting  will  a  room  contain 
which  (being  girt  over  the  mouldings)  is  16  feet  6  inches, 
and  the  compass  of  the  room  97  feet  6  inches  ? 

Operation. 

By  decimals.  By  duodecimals. 

Ft.  in.  97     6 

97     6=97,5  16     6 

16     6=16,5 


4875 
5850 
975 


-r  9  )  1608,75  feet  -r-9  )  1608 


Yards  178,6,75  178,6,9 

Ans.  178J  yards. 

2.  A  gentleman  had  a  room  painted  at  8J  cents  per  yard, 
the  measure  whereof  is  as  follows,  viz.  the  height  11  feet 

7  inches,  the  compass  74  feet  10  inches,  the  door  7  feet  6 
inches  by  3  feet  9  inches ;  five  window  shutters,  each  6  feet 

8  inches  by  3  feet  4  inches;  the  breaks  in  the  windows  14 
inches  deep,  and  8  feet  high ;  the  opening  for  the  chimney  6 
feel  -    inches  by  5  feet,  to  be  deducted,  the  shutters  and  doors 
are  painted  on  both  sides ;  what  will  the  whole  come  to  ? 

Ans.  10  dolls.  43  cts. 

3.  How  many  yards  of  painting  are  there  in  a  room,  the 
length  whereof  is  20  feet,  its  breadth  14  feet  6  inches,  and 
height  10  feet  4  inches;  deducting  a  fire-place  of  4  feet  by 
4  feet  4  inches,  and  two  windows,  -  each  6  feet  by  3  feet  2 
inches?  Arts.  73^T  yards. 


180  GLAZIERS'  WORK. 

4.  What  cost  the  painting  of  a  room  at  6  cents  per  yard , 
its  length  being  24  feet  6  inches,  its  breadth  16  feet  3  inches, 
and  height  12  feet  9  inches ;  also  the  door  is  7  feet  by  3 
feet  6  inches,  and  the  window  shutters  of  two  windows, 
each  7  feet  9  inches  by  3  feet  *6  inches,  but  the  breaks  of  the 
windows  themselves,  are  8  feet  6  inches  high,  and  1  foot  3 
inches  deep ;  deducting  a  fire-place  cf  5  feet  by  5  feet  6 
inches.  A?is.  7  dolls.  66  cts.  9£  m. 


SECTION  10. 

OF  GLAZIERS'  WORK. 

GLAZIEKS  compute  their  work  in  square  feet ;  and  the  di- 
mensions are  taken  either  in  feet,  inches,  and  seconds,  &c. 
or  in  feet,  tenths,  hundredths. 

Windows  are  sometimes  measured  by  taking  the  dimen 
sions  of  one  pane,  and  multiplying  its  .superficies  by  the 
number  of  panes.  But  more  generally  they  measure  the 
length  and  breadth  of  the  window  over  all  the  panes,  and 
their  frames  for  the  length  and  breadth  of  the  glazing.  And 
oftentimes  the  work  is  estimated  at  so  much  per  pane  ac- 
cording to  the  size. 

Circular,  or  ovtil  windows,  as  fan-lights,  &c.  are  measured 
as  if  they  were  square,  taking  for  their  dimensions  the 
greatest  length  and  breadth,  as  a  compensation  for  the  waste 
of  glass,  and  labor  in  cutting  it  to  the  proper  forms. 

EXAMPLES. 

1.  I  low  many  square  feet  are  contained  in  a  window, 
which  is  4  feet  3  inches  long,  and  2  feet  9  inches  broad? 

Hv  decimals.  By  duodecimals. 

Ft.  in."  Ft.    in. 

4     «  =  4, 25  the  length  4       3 

2      9  -=2,75  the  breadth  2        9 


2125  8        6 

2975  3        2 

sf,0  


11/H75  foot. 


11        S       3  Ans. 


MEASUREMENT  OF  GROUJND  181 

2.  If  a  window  be  7  feet  3  inches  high,  and  3  feet  5  inches 
broad,    how    many  square  feet   of   glazing   are   contained 
therein ?  .  An*.  24  feet  9  inches. 

3.  There  is  a  house  with  three  tiers  of  windows,  7  in  a 
tier;  the  height  of  the  first  tier  is  6  feet  11  inches,  of  the 
second,  5  feet  4  inches,  and  of  the  third,  4  feet  3  inches ;  the 
breadth  of  each  window  is  3  feet  6  inches :  what  will  the 
glazing  come  to  at  14^  cents  per  foot? 

Ans.  58  dolls.  61  cts. 

4.  What  will  the  glazing  of  a  triangular  sky-light  come 
to  at  10  cents  per  foot,  the  base  being  12  feet  6  inches  long, 
and  the  perpendicular  height  16  feet  9  inches? 

Ans.  10  dolls.  465  cts. 

5.  What  is  the  area  of  an  elliptical  fan-light  of  14  feet 
0  inches  in  length,  and  4  feet  9  inches  in  breadth  ? 

Ans.  68  feet  10  inches. 

6.  There  is  a  house  with  three  tiers  of  windows,  and  & 
in  each  tier;  the  height  of  the  first  tier  is  7  feet  10  inches, 
of  the  second,  6  feet  8  inches,  of  the  third  £  fret  4  inches 
and  the  common  breadth  3  feet  11  inche?  ,   what  will  the 
glazing  come  to  at  14  cents  per  foot? 

Ans.  &•'  <Kis.  87  j  cts. 


SECTION  11 
MEASUREMENT  OF  GKOUND. 

1st.    To  find  the  content  of  a  square  piece  of  ground. 

RULE. 

MULTIPLY  the  base  in  perches,  yards  or  feet,  as  the  case 
may  be,  by  the  perpendicular,  and  the  product  will  be  the 
answer  required. 

Note.  1.  Any  area,  or  content  in  perches,  being  divided  by  160,  will 
give  the  content  in  acres ;  the  remaining-  perches,  if  more  than  40, 
being  divided  by  40,  will  give  the  roods,  or  quarter  acres,  and  the  last 
remainder,  if  any,  will  be  perches. 

2.  Ground  is  generally  measured  by  chains,  of  two  poles  or  rods  in 
length  ;  the  two  pole  chain  measures  33  feet.  Chains  of  4  poles  are 
sometimes  used,  and  sometimes  chains  or  poles  of  one  rod  in  length 
only. 

Q 


182  MEASUREMENT  OF  GROUND 

EXAMPLE. 

1.  In  a  square  field,  A,  B,  C,  D,  each  side  of    which 
measures  40  rods,  or  poles,  how  many  acres  ? 

Operation. 

40  DC 

40 


4,0)  160,0 


10  Acres. 


4)40  A      40       B 

10  acres.  Ans. 

2.  In  a  square  field,  each  side  of  which  measures  35  two 
pole  chains,  how  many  acres? 

Ans.  30  acres  2  roods  20  perches. 

8.  A  piece  of  square  ground  measures  16^  perches  on 
each  side ;  what  is  the  content  in  acres  ? 

Ans.  1  acre  2  roods  32|  perches. 

2d.   Tojind  the  content  of  an  oblong  square  piece  of  ground, 
called  a  parallelogram. 

RULE. 

Multiply  the  length  by  the  breadth,  and  the  product  will 
be  the  answer. 

EXAMPLE. 

1.  There  is  an  oblong  square  piece  of  ground,  A,  B,  C,  D, 
the  longest  sides  of  which  measure  64  perches,  and  the 
shortest  sides,  or  ends,  measure  40 ;  how  many  acres  does 
it  contain  ? 

Operation.  D  C 

64= the  length 
40= the  breadth 


4,0 )  256,0  perches 
4)64 


16  Acres. 


64 


16  acres.   An*. 


MEASUREMENT  oV  GROUND.  183 

2.  In  a  piece  of  ground  lying  in  the  form  of  an  oblong 
square,  the  length  measures  120  perches,  and  the  breadth 
84;  what  is  its  content  in  acres?  Ans.  63  acres. 

3.  A  lot  of  ground  lying  in  the  form  of  an  oblong  square, 
measures  240  feet  in  length,  and  120  in  breadth;  what  is 
its  content  in  acres '.' 

Ans.  0  acres  2  quarters  25  perches  21 3J  feet. 

4.  There  is  an  oblong  piece  of  ground,  whose  length  is 
14  two  pole  chains  25  links,  and  breadth  8  chains  37  links  ; 
how  many  acres  does  it  contain  ? 

Ch.  L.    Perches. 

8     37  =  17,48  breadth 
14     25=      29  length 

15732 
3496 

4,0)50^,92 
4)12^26" 


3    0     26,92 

Ans.  3  acres  0  quarters  27  perches  nearly. 
Note.  The  English  statute  perch  is  5£  yards,  the  two  pole  chain  is 
11  yards,  or  33  feet,  and  is  divided  into  50  links;  the  four  pole  chain 
is  22  yards,  or  66  feet,  and  contains  100  links ;  hence  the  length  of  a 
link  in  a  statute  chain  is  7,92  inches,  and  25  links  make  1  rod.  And 
consequently,  if  the  links  be  multiplied  by  4,  carrying  1  to  the  chains 
for  every  25  links,  and  the  chains  by  2,  the  product  will  be  perches, 
and  decimals  of  a  perch. 

5.  An  oblong  piece  of  ground   measures   17   two  pole 
chains  and  21  links  in  length,  and   15  chains  38  links  in 
breadth  ;  how  many  acres  are  contained  therein  ? 
Ch.    L.  Ch.    L. 

17     21  15     38 

24  24 


34     84  perches,  the  length.         31     52  p.  the  breadth. 
Then  34,84x31,52=1098,1568  perches=6  acres  3  qr. 
18,1 5  -f-  perches. 
3d.   To  find  the  content  of  a  triangular  piece  of  ground. 

RULE. 

Multiply  the  base  by  half  the  perpendicular,  or  the  per- 
pendicular by  half  the  base,  or  take  half  the  product  of  the 
base  into  the  perpendicular. 


184 


MEASUREMENT  OF  GROUND 


B  D  C 

1.  Let  A,  B,  C,  be  a  triangular  piece  of  ground,  the 
ongest  si«o  or  base  B,  C,  is  24  chains  38  links,  and  perpen- 
dicular, A  D,  13  chains  29  links;  how  many  acres  does  it 
contain  ? 

Operation. 
Ch.     L. 

24     38=49,52  perches 
13     28=27,12 


9904 
4952 
34664 
9904 

1342,9824 

Half  the  sum  is  4,0)67,1,4912  perches 
4  )  16,31 


4  0  31,4 
Ans.  4  acres  0  roods  31,4  perches. 

2.  In  a  triangular  piece  of  ground,  the  base  or  longest 
side  measures  75  perches,  and  the  perpendicular  50 ;  how 
many  acres  does  it  contain  ? 

Ans.  11  acres  2  qrs.  35  perches. 

3.  How  much  will  a  triangular  piece  of  ground  come  to 
at  45  dollars  per  acre,  the  longest  side  or  base  of  which 
measures  120  perches,  and  the  perpendicular  84  perches. 

Ans.  1417  dolls.  50  cents. 

4.  How  many  superficial  yards  are  contained  in  a  trian- 
gular piece  of  ground,  fho  bavc  of    vhirh  measures  140  feet 
and  the  perpendicular  70  feet'!  Ans   544  vards  4  feet. 


MEASUREMENT  OF  GROUND. 


185 


4ith.    To  find  the  content  of  a  piece  of  ground,  in  the  form 
of  an  oblique  parallelogram. 

RULE. 

Multiply  the  base  into  the  perpendicular  height  for  the 
content. 

EXAMPLE. 
D  C 


A       E  B 

.  Let  A,  B,  C,  D,  be  a  piece  of  ground  in  the  form  of 
t~n  oblique  parallelogram,  the  base  of  which,  A,  B,  measures 
44  perches,  and  the  perpendicular,  D,  E,  40  perches ;  how 
many  acres  does  it  contain  1 

44  length 
-40  breadth 


4,0)176,0  perches 


4)    44 

11  acres.     Ans.  11  acres. 

2.  A  piece  of  ground  lying  in  the  form  of  an  oblique  par- 
allelogram, is  found  to  measure  80  perches  along  its  base, 
and  its  perpendicular  height  24  perches;  how  many  acres 
does  it  contain?  Ans.  12  acres. 

5th.    To  find  the  content  of  a  piece  of  ground  bounded  by 
four  sides,  none  of  which  are  parallel  or  equal. 

RULE. 

Find  the  length  of  a  diagonal  line  between  the  two  most 
distant  corners,  and  multiply  this  line   by  the  -  sum  of  the 
two  perpendiculars  falling  from  the  other  corners  to  that  di- 
agonal line,  and  half  the  product  will  be  the  area. 
Q2 


186 


MEASUREMENT  OF  GROUND. 


1.  Let  A,  B,  C,  D,  be  a  field  with  four  irregular  and  un- 
equal sides,  the  diagonal  line  of  which,  A,  C,  measures  80 
perches,  the  perpendicular,  B,  m,  measures  25  perches,  and 
the  other  perpendicular,  D,  n,  35  perches;  how  many  acres 
does  it  contain  ? 

80  the  length  of  the  diagonal  line. 

25 -f  35=60  the  sum  of  the  two  perpendiculars. 

2 ) 4800 


4,0)240,0  perches 


4)    60 

15  acres.  An*. 

2.  In  a  field  of  four  unequal  sides,  the  diagonal  line  be- 
tween the  two  most  distant  corners  measures  120  rods,  and 
the  perpendiculars  measure,  the  one  48,  and  the  other  24 
rods ;  required  the  number  of  acres  it  contains? 

Ans.  27  acres. 

6/7i.    To  find  the  area  of  a  pitce  of  ground  lying  in  a  cir- 
cle y  or  ellipsis. 

RULE. 

Multiply  the  square  of  the  circle's  diameter,  or  the  pro- 
duet  of  the  longest  and  shortest  diameters  of  the  ellipsis,  by 
the  derimal  number  ,7854,  the  product,  will  give  the  area. 

Note.    In  any  circle,  the 
Diameter  multiplied 
Circumference  flivid'-il 


MEASUREMKiVr  OF  GROUND.  187 

EXAMPLE. 

1.  How  many  acres  are  contained  in  a  circular  piece  of 
ground,  whose  diameter  measures  320  perches,  or  1  mile  ? 
320X320=102400 

,7854 


409600 
512000 
819200 
716800 


4,0 )  80424,9600  perches 
4)2010,24,9 


502  2  24,9 
Ans.  502  acres  2  qr.  24,9  perches. 

2.  A  gentleman  has  an  elliptical  -yard  in  front  of  his 
house,  the  longest  diameter  of  which  measures  30  perches, 
and  the  shortest  20  ;  how  much  ground  is  contained  therein  ? 

Ans.  2  acres  3  qr.  31,2  perches. 

3.  How  many  square  yards  are  contained  in  a  circular 
piece  of  ground,  the  diameter  of  which  measures  160  feet? 

Ans.  22 34  + yards. 

From  the  foregoing  simple  methods  of  finding  the  con- 
tents of  ground  lying  in  different  forms,  it  will  readily  be 
seen,  that  the  content  of  fields  and  small  pieces '  of  land, 
lying  in  any  shape  whatever,  and  bounded  by  any  number 
of  sides,  may  be  calculated,  without  having  recourse  to  the 
more  expensive  and  troublesome  practice  of  employing  a 
regular  surveyor.  No  other  apparatus  than  a  common  rod- 
pole,  or  line  of  a  known  length,  is  requisite.  Pieces  of  land 
having  more  than  four  boundary  lines,  may  be  easily  divided 
into  squares,  parallelograms,  triangles,  (fee.  and  each  calcu 
iated  separately  by  some  of  the  foregoing  rules,  and  then  the 
whole  amount  added  into  one  sum  for  the  content.  It  is  of 
great  importance  to  every  practical  farmer  to  know  the  size 
of  the  different  fields  which  he  cultivates.  Besides  the  satis- 
faction thereof,  this  knowledge  is  necessary  to  enable  him  to 
regulate  the  quantity  of  seed  which  he  should  sow,  as  well 
the  price  for  clearing^  plowing,  planting,  reaping,  &c. 


188  GAUGING. 

SECTION  12. 

OF  GAUGING. 

GAUGING  is  taking  the  dimensions  of  a  cask  in  inches,  to 
find  its  content  in  gallons. 

RULE. 

1.  Find  the  mean  diameter,  between  the  head  and  bung 
diameters,  by  adding  two-thirds  of  the  difference  between 
them  to  the  head  diameter.     If  the  staves  be  but  little  curv- 
ing from  the  head  to  the  bung,  add  only  six-tenths  of  this 
difference. 

2.  Square  the  mean  diameter,  so  found,  and  multiply  the 
product  by  the  length  of  the  cask  in  inches,  for  the  content 
thereof  in  cubic  inches. 

3.  Divide  the  cubic  inches,  so  found,  by  294,  for  wine  or 
spirits,  and  by  359  for  ale ;  the  quot'ent  will  be  the  answer 
in  gallons. 

EXAMPLE. 

1.  How  many  gallons  of  wine  will  a  cask  contain  whose 
bung  diameter  is  31  inches,  head  diameter  25  inches,  and 
whose  length  is  3  feet,  or  36  inches  1 

Operation. 

31  bung  diam.  25  head  diameter 

25  head  diam.         f  of  6=   4  two-thirds  difference 

6  difference.  29 

29 

261 

58 

841  square  of  the  mean  diam. 
36  the  length 


5046 
2523 

30276  cubic  inches. 
Then  80276 -f- 294  =102ffJ  gals.  Or,  102  gals.  3  qt. 


GAUGING.  189 

2.  The  diameter  of  a  barrel  at  the   bung   measures  24 
inches,   and  at  the  head   18   inches,  and   its  length  is  24 
inches  ;  what  is  its  content  in  wine  measure  1 

Ans.  39f  f  gals. 

3.  How  many  gallons  of  spirits  will    a   cask   contain, 
whose  bung  diameter  is  36  inches,  head  diameter  28  inches, 
and  whose  length  is  3  feet  4  inches  ? 

Ant.  ISlfN,  gals. 


4.  What  is  the  content,  in  ale  measure,  of  a  barrel  whose 
bung  diameter  measures  18  inches,  head  diameter  15  inches, 
and  whose  length  is  2  feet  5  inches  ? 

Ans.  23i||.  gals. 

5.  Bought  a  barrel  of  ale  of  the  following  dimensions, 
viz.  bung  diameter  22  inches,  head  diameter  18  inches,  and 
length  3  feet ;  how  many  gallons  does  it  contain  ? 

Ans.  42|f|  gals. 

Of  the  Gauging  or  Diagonal  Rod. 

The  diagonal  rod  is  a  square  rule,  having  four  faces,  be- 
ing commonly  four  feet  long,  and  folding  together  by  joints. 
This  instrument  is  used  for  gauging,  or  measuring  casks, 
and  computing  their  contents,  and  that  from  one  dimension 
only,  namely,  the  diagonal  of  the  cask  ;  that  is,  from  the 
middle  of  the  bung- hole,  to  the  meeting  of  the  head  of  the 
cask,  with  the  stave  opposite  to  the  bung ;  being  the  longest 
line  that  can  be  drawn  within  the  cask  from  the  middle  of 
the  bung-hole.  Am!  accordingly  one  face  of  the  rule  is  a 
scale  of  inches,  for  measuring  this  diagonal,  to  which  are 
placed  the  areas  in  ale  gallons,  of  circles  to  the  correspond- 
ing diameters,  in  like  manner  as  the  lines  on  the  under  sides 
of  the  three  slides,  in  the  sliding  rule.  On  the  opposite  face 
are  two  scales  of  ale  and  wine  gallons,  expressing  the  con- 
tents of  casks  having  the  corresponding  diagonals.  And 
these  are  the  lines  which  chiefly  form  the  difference  between 
this  instrument  and  the  sliding  rule. 

EXAMPLE. 

The  rod  being  applied  within   the  cask  at  the  bung-hole. 


190  MECHANICAL  POWERS. 

the  diagonal  was  found  to  be  34,4  inches  ;  required  the  con- 
tent in  gallons. 

Now,  to  34,4  inches,  will  be  found  corresponding  on  the 
rod,  90 1  ale  gallons,  and  111  wine  gallons,  the  content  re- 
quired. 

Not e.  In  taking  the  length  of  a  cask  to  find  the  cubic  inches,  an 
allowance  must  be  made  for  the  thickness  of  both  the  heads  of  1  inch, 
of  1^  inch,  or  2  inches,  according  to  the  size  of  the  cask ;  and  the  head 
diameter  must  always  be  taken  close  to  the  chime.  The  contents  ex- 
hibited by  the  rod,  answer  only  to  casks  of  the  common  form. 


SECTION  13. 

OF  MECHANICAL  POWERS. 

1st.    OF    THE    LEVER. 

To  find  what  weight  may  be  raised  or  balanced  by  any 
given  power. 

RULE. 

As  the  distance  between  the  body  to  be  raised,  and  ful- 
crum, or  prop, 

Is  to  the  distance  between  the  prop,  and  the  point  where 
the  power  is  applied, 

So  is  the  power  to  the  weight  which  it  will  raise. 

EXAMPLE. 

1.  If  a  man  weighing  150  Ib.  rest  on  the  end  of  a  levei 
12  feet  long ;  what  weight  will  he  bafemce  on  the  other  end, 
supposing  the  prop  1  £  foot  from  the  weight  ? 

Operation. 

12= the  length  of  the  lever 
l,5=distance  of  the  weight  from  the  prop 


10,5= the  distance  from  the  prop  to  the  man. 
Then,  as  1,5  :  10,5  : :   150  :  1050.     Ans. 
2.  The  pea  of  a  pair  of  steelyards  weighing  5  Ib.  is  re- 
moved 20  inches  back  from  the  fulcrum  ;  what  weight  will 


MECHANICAL  TOWERS.  1^1 

it  balance,  suspended  at  I  irvh  distance  on  the  opposite  si  le? 

Arts  100  lo. 

*2d.     OF    THE    WHEEL    AND    AXLE. 

To  find  what  power  must  be  applied  at  the  wheel,  to  r<  \ise 
a  given  weight  suspended  to  the  axle  ;  or  what  weight  ai 
the  axle  will  be  raised  by  a  given  power  at  the  wheel. 

RULE. 

As  the  diameter  of  the  axle  :  is  to  the  diameter  of  th< 
wheel  : :  so  is  the  power  applied  to  the  wheel  :  to  the  weight 
suspended  to  the  axle. 

EXAMPLE. 

1.  It  is  required  to  make  a  windlass   in  such  a  manner, 
that  1  Ib.  applied  to  the  wheel,  shall   be  equal  to  12  lb.  sus- 
pended to  the  axle;  now  allowing  the  axle  to  be  4  inches 
diameter,  what  must  be  the  diameter  of  the  wheel  ? 

lb.  in.      lb.     in. 
As  1  :  4  : :  12  :  48  =  4  feet  the  diameter  of  the  wheel.  Ans. 

2.  Suppose  the  diameter  of  an  axle  to  be  6  inches,  and 
that  of  the  wheel  5  feet;  what  power  at  the  wheel  will  bal- 
ance 10  lb.  at  the  axle?  Ans.  1  lb. 

3d.    OF    THE    SCREW. 

In  the  screw  there  are  four  things  to  be  considered :  viz. 
the  power,  the  weight,  the  distance  between  the  threads,  and 
the  circumference.  To  find  any  one  of  these,  the  other 
three  being  given,  observe  the  following  proportional 

RULE. 

As  the  distance  between  the  threads  of  the  screw : 
Is  to  the  circumference  : : 
So  is  the  power  : 
To  the  weight. 

Note.  1.  To  find  the  circumference  of  the  circle  described  by  the 
end  oi  the  lever ;  multiply  the  double  of  the  lever  by  3,14150,  and  the 
product  will  be  the  circumference. 

2    It  is  usual  to  abate  £  of  the  effect  of  the  machine  for  the  friction 


192  PROMISCUOUS  QUESTIONS. 

EXAMPLE. 

There  is  a  screw  whose  threads  are  an  inch  asunder ;  the 
lever  by  which  it  is  turned  is  36  inches  long,  and  the  weight 
to  be  raised  a  ton,  or  2240  Ib. ;  what  power  or  force  must 
be  applied  to  the  end  of  the  lever  sufficient  to  turn  the  screw 
that  is  to  raise  this  weight  ? 

Thus,  the  lever  36  X  2=72,and72  X  3,14159^226,194  + 
the  circumference. 

Circum.     in.        Ib.  Ib. 

Then,  as  226,194  :  1   ::  2240  :  9,903  the  power.   Ans. 


A  COLLECTION  OF  PROMISCUOUS  QUESTIONS,  TO  EXERCISE 
THE  SCHOLAR  ON  THE  FOREGOING  RULES. 

1.  What  is  the  sum  of  2578,  added  to  itself? 

Ans.  5156. 

2.  What  is  the  difference  between  14676,  and  the  fourth 
of  itself!  Ans.  11007. 

3.  There  is  the  sum  of  1468  dollars  in  three  bags;  the 
first  contains  461,  the  second  581,  how  many  are  in  the 
third  bag?  Ans.  426. 

4.  What  is  the  sum  of  the  third  and  half  third  of  1  dol- 
lar ?  Ans.  50  cts. 

5.  What  number  is  that  which  being  multiplied  by  45  the 
product  will  be  1080?  Ans.  24. 

6.  Required  the  quotient  of  the  square  of  476,  divided 
by  the  half  of  itself,  or  its  single  power?  Ans.  952. 

7.  A  general  drawing  up  his  army  into  a  solid  square, 
found  he  had  231  over  and  above,  but  increasing  each  side 
with  one  soldier,  he  wanted  44  to  complete  the  square ;  how 
many  men  did  his  army  consist  of?  Ans.  19000. 

8.  What  number  added  to  the  cube  of  21,  will  make  tin- 
sum  equal  to  113  times  147  ?  Ans.  7350. 

9.  A  person  possessed  of  $  of  a  ship,  sold  f  of  his  share 
for  1260  dollars;  what  was  the  value  of  the  whole  ship  at 
Th"  sani"  rate?  Ans.  5040  dolls. 

10.  A  jruJirdinn    paid  his  ward  3500  dollars  for  2500  dol- 
lars, uhieh  he   had  in  his  hands  for  8  years;   what   rate  of 

fiU'rest  did  lie  allow  him7  .••!//*.  5  per  cent. 


PROMISCUOUS  QUESTIONS.  193 

11.  A  young  man  received  210  dollars,  which  was  f  of 
his  elder  brother's  portion ;  now  three  times  the  elder  bro- 
ther's portion  was  half  of  the  father's  estate ;  how  much  was 
the  estate  worth?  Ans.  1890  dolls. 

12.  A  broker  bought  for  his  principal  in  the  year  1720, 
the  sum  of  400  dollars  capital  stock,  in  the  south  sea,  at  650 
per  cent,  and  sold  it  again  when  it  was  worth  but  130  dollars 
per  cent. ;  how  much  was  lost  upon  the  whole  ? 

Ans.  2080  dolls. 

13.  A  gentleman  went  to  sea  at  17  years  of  age;  8  years 
after  he  had  a  son  born,  who  lived  40  years,  and  died  before 
his  fether ;  after  whom  the  father  lived  twice  20  years,  and 
then  died  also ;  I  demand  the  age  of  the  father  when  he 
died?  Ans.  Ill  years. 

14.  A,  B,  and  C,  entered  into  partnership  in  trade,  A  put 
in  a  sum  unknown,  B  put  in  20  pieces  of  cloth,  and  C  put 
in  500  dollars ;  at  the  end  of  one  year  they  had  gained  1000 
dollars,  whereof  A  received  350  dollars  for  his  share,  and  B 
400  dollars  ;  required  C's  share,  how  much  A  put  in,  and  the 
value  of  B's  cloth? 

Ans.  C's  share  250  dollars, — A  put  in  700  dollars, 
— B's  cloth  was  worth  800  dollars. 

15.  A  captain  and  160  sailors  took  a  prize  worth  2720 
dollars,  of  which  the  captain  gets  J  for  his  share,  and  the 
rest  is  equally  divided  among  the  sailors  ;  what  was  each 
one's  part  ? 

Ans.  The  captain  gets  544  dollars,  and  each  sailor 
13  dollars  60  cents. 

16.  A  lady  tells  her  husband,  upon  her  marriage,  that  her 
fortune,  the  interest  of  which  for  one  year  at  6  per  cent, 
was  972  dollars,  was  but  the  f  of  the  interest  of  her  father's 
estate  for  three  years,  at  the  same  rate  per  cent. ;  what  was 
the  lady's  fortune,  and  what  was  the  value  of  her  father's  es- 
tate?" 

Ans.  Her  fortune  was  16,200  dollars,  and  her  father' 
estate  was  150,000  dollars. 

17.  A  stone  measures  4  feet  6  inches  long,  2  feet  9  inches 
broad,  and  3  feet  4  inches  deep ;  how  many  cubic  feet  does 
it  contain  ?  Ans.  41  feet  3  inches. 

18.  Suppose  I  of  a' mast  or  pole  stands  in  the  ground, 
12  feet  in  tne  water,  and  f  of  its  length  above  the  water; 
what  is  its  whole  length?  Ans.  216  feet. 

R 


194  PROMISCUOUS  QUESTIONS. 

19.  A  gentleman   being   asked  his  age,  answered,  my 
grandfather  is  112  years  old,  and  my  father  4  of  his  age, 
whilst  mine  is  but  ^  of  my  father's ;  what  was  his  age  ? 

Ans.  21 J  years. 

20.  A  person  who  was  possessed  of  f  share  of  a  copper 
mine,  sold  3  of  his  interest  therein  for  1710  dollars;  what 
was  the  value  of  the  property  at  the  same  rate  ? 

Ans.  3800  dollars. 

21.  There  are  two  numbers,  the  one  63,  the  other  half  as 
much ;  required  the  product  of  their  squares,  and  the  differ- 
ence of  their  product  and  sum  1 

'    A       $  Product  of  the  squares  3938240,25. 
5*  J  Difference  1890. 

22.  Two  men  set  out  at  the  same  time  from  the  same 
place,  but  go  contrary  ways,  and  each  of  them  travels  34 
miles  a  day ;  required  the  time  in  which  they  will  have 
travelled  2000  miles  ?  Ans.  29  days  9  hours  52ff  mi. 

23.  If  a  cannon  may  be  discharged  twice  with  6  Ib.  of 
powder,  how  many  times  will  7Cwt.  3qr.  I7lb.  discharge  the 
same  piece  ?  Ans.  295  times. 

24.  What  number  is  that,  to  which  if  you  add  f  of  itself, 
the  sum  will  be  20?  Ans.  12. 

25.  What  number  is  that,  which  being  divided  by  j},  the 
quotient  will  be  21  ?  A?is.  15J. 

26.  What  number  is  that,  which  being  multiplied  by  15, 
the  product  will  be  \ 1  Ans.  -^\ 

27.  WThat  number  is  that,  from  which  if  you  take  f ,  the 
remainder  will  be  J?  Ans.  ff 

28.  A  gentleman  wishing  to  distribute  some  money  among 
a  number  of  children,  found  he  wanted  8  cents  to  give  them 
:J  cents  a  piece,  he  therefore  gave  each  2  cents,  and  had 
three  cents  left;  how  many  children  were  there?  Ans.  11. 

29.  In  what  time  will  500  dollars  amount  to  1000,  at  6 
per  cent,  per  annum?  Ans.  16  years  8  months. 

30.  When  \  of  the  members  of  congress  were  assembled 
-t  15,  there  were  J-flO  absent;  how  many  members  were 
in  all?  Ans.  150. 

31.  If  the  earth  be  360  degrees  round,  each  69 £  miles, 
how  long  would  it  take  a  man  to  travel  orce  round,  at  20 
miles  a  day,  admitting  there  were  no  obstacles  in  the  way, 
and  reckoning  365  i  days  in  the  year. 

Ans.  3  years  155J  days. 


PROMISCUOUS  QUESTIONS.  195 

32.  -What  is  the  mean  time  for  paying  100  dollars  a;  3| 
months,  150  dollars  at  4^  months,  and  204  dollars  at  5if 
months  ?  Ans.  4  months  23Jf  A  days. 

33.  If  A  can  do  a  piece  of  work  alone  in  7  days,  and  B 
do  the  same  in  12,  how  long  will  it  require  them  both  to- 
gether  ?  Ans.  4TV  days. 

34.  A  minor  of  14  years  of  age,  had  an  annuity  left  him 
of  400  dollars  ;  this  sum   his  guardian  agreed  to  receive 
yearly,   and  allow  him  compound  interest  at  5  per  cent, 
thereon,  till  he  should  arrive  at  21  years  of  age ;  how  much 
must  he  then  receive  ?  Ans.  3256  dolls.  80  -f  cents. 

35.  Sold  goods  to  the  amount  of  700  dollars  for   four 
months  ;  what  was  the  present  worth,  at  5  per  cent,  simple 
interest?  Ans.  688  dolls.  52 -f  cents. 

36.  Three  persons,  A,  B,  and  C,  purchased  a  lot  in  part- 
nership, for  which  A  advanced  f ,  B  -f ,  and  C  140  dollars  ; 
what  sum  did  A  and  B  pay,  and  what  part  of  the  lot  be- 
longed to  C  ? 

C  A  paid  267  dolls.  27  +  cts. 
Ans.   <  B  paid  305    —     45J    — 
(  and  C  had  J-J  parts. 

37.  A  gentleman    finding   several    beggars   at  his  door, 
gave  to  each  four  cents,  and  had  sixteen  left ;  but  if  he  had 
given  to  each  six  cents,  he  would  have  wanted  12  ;  how 
many  beggars  were  there  ?  Ans.  14. 

38.  B  and  C  can  build  a  wall  in  18  days,  but  with  the 
assistance  of  A  they  can  do  it  in  11  days;  in  what  time 
can  A  do  it  alone  1 

Suppose  the  work  to  consist  of  198  parts. 

Then  198-^18=11  parts  performed  by  B  and  C,  in  one 
day. 

Again,  198-^11  =  18,  performed  by  A,  B,  and  C,  in  one 
day. 

But  18 — 11=7  parts  performed  by  A  alone. 
P.        D.  P.  D.  h.  m. 

And  as  7     :     1     : :     198     :     28    3   25f     Ans. 

39.  Twenty  members  of  congress,  30  merchants,  24  law- 
yers, and  24  citizens,  spent  at  a  dinner  192  dollars  ;  which 
sum  was  divided  among  them  in  such  a  manner,  that  4 
members  of  congress  paid  as  much  as  5  merchants,  10 
merchants  as  much  as  16  lawyers,  and  8  lawyers  as  much 


196          PROMISCUOUS  QUESTIONS. 

as  12  citizens;  the  question  is  to  know  the  sum  of. money 
paid  by  all  the  members  of  congress ;  also,  by  the  mer- 
chants, lawyers,  and  citizens  ? 

Ans.  The  20  members  of  congress  paid  60  dollars 

the  30  merchants  paid  72,  the  24  lawyers  paid  36 

and  the  24  citizens  paid  24. 

40.  What  difference  is  there  between  a  piece  of  ground 
28  perches  long,  by  20  broad,  and  two  others  each  of  half 
those  dimensions  ?  Ans.  1  acre  3  qrs. 

41.  Required  the  dimensions  of  a  parallelogram,  con- 
taining 200  acres,  which  is  40  perches  longer  than  wide  ? 

Ans.  200  perches  by  160, 

42.  How  many  acres  are  contained  in  a  square  field,  the 
diagonal  of  which  is  20  perches  more  than  either  of  its 
sides?  Ans.  14  acres  2  qrs.  11  per. 

43.  The  paving  of  a  triangular  yard,  at  I8d.  per  foot, 
came  to  100Z. ;  the  longest  of  the  three  sides  was  88  feet ; 
what  then  was  the  sum  of  the  other  two  equal  sides  1 

Ans.  106,85  feet. 

44.  Required  the  length  of  a  line  by  which  a  circle  that 
shail  contain  just  half  an  acre  may  be  laid  off? 

Ans.  21%  yards. 

45.  A  ceiling  contains  114  yards  6  feet  of  plastering,  and 
the  room  is  28  feet  broad  ;  what  is  its  length  ? 

Ans.  36f  feet. 

46.  A  common  joist  is  7  inches  deep,  and  2£  thick,  but  I 
want  another  just  as  big  again,  that  shall  be  three  inches 
thick  ;  what  must  be  its^ether  dimensions  ? 

Ans.  llf  inches. 

47.  If  20  feet  of  iron  railing  weigh  half  a  ton,  when  the 
oars  are  an  inch  and  a  quarter  square,  what  will  50  feet 
come  to  at  3 \d.  per  pound,  the  bars  being  but  £  of  an  inch 
square  ?  Ans.  20Z.  Os.  2d. 

43.  A  may-pole  whose  top  being  broke  off  by  a  blast  of 
\vind,  struck  the  ground  at  15  feet  distance  from  the  foot  of 
the  pole ;  what  was  its  whole  height,  supposing  the  length 
of  the  broken  piece  to  be  39  feet  ?  Ans.  75  ft. 

49.  Required  a  number,  from  which  if  7  be  subtracted,  arid 
the  remainder  be  divided  by  8,  and  the  quotient  be  multiplied 
by  5,  and  4  added  to  the  product,  the  square  root  of  the  sum 
extracted,  and  three-fourths  of  that  root  cubed,  the  cube  di- 
vided by  9,  the  last  quotient  will  be  24  ?  Ans.  103. 


,          PROMISCUOUS  QUESTIONS.  197 

50.  A  vintner  has  a  cask  of  wine  containing  500  galls,  of 
which  he  draws  50  galls,  and  fills  it  up  with  water,  and  re- 
peats the  same  thing  five  times ;  I  demand  what  quantity  of 
wine,  and  also  of  water,  is  then  in  the  cask  ? 

Ans.  295  galls.  1  qt.  of  wine,  and  204  galls.  3  qts. 
of  water  nearly. 

51.  Since  a  pile  of  wood  4  feet  long,  4  feet  high,  and  8 
feet  broad,  makes  a  cord,  what  part  of  a  cord  will  be  in  a 
pile  of  half  the  dimensions  each  way  ?  Ans.  %  part. 

The  answers  to  the  following  questions  are  designedly  omitted,  that 
the  scholar  may  be  induced  to  apply  to  the  resources  of  his  own  mind 
alone  for  the  solution  thereof.  Without  habits  of  reflection  and  inves- 
tigation are  acquired,  by  which  he  can  compare,  examine  and  apply 
the  various  rules  and  directions  that  are  contained  in  this  treatise,  he 
never  can  have  any  good  claim  to  be  considered  a  proficient  in  arithmetic 

52.  A  owed  B  1864  dollars,  for  which  he  gave  his  note, 
on  interest,  bearing  date  April  1st,  1817. 

On  the  back  of  the  note  are  the  following  endorsements*, 
viz. 

Oct.  15th,  1817.    Received  in  cash  225  dolls.  50  cts. 

Jaiv  10th,  1818.    Received  in  cash  150    — 

Same  date,  one  bag  of  coffee ;  weight  1  Cwt.  22lb.  at  29 
cents  per  pound. 

May  16th.    Received  3  ton  of  iron  at  195  dolls,  per  ton. 

What  is  the  sum  due  from  A  to  B,  on  the  1st  of  August 
1818? 

53.  How  many  cords  are  there  in  a  pile  of  wood  36  feet 
long,  6|  feet  wide,  and  8J  feet  high? 

54.  If  a  man  spends  356  dollars  34  cgnts  per  year,  how 
much  will  it  be  per  day  ? 

55.  A  bankrupt,  whose  whole  property  is  worth  2564 
dollars  95 1  cents,  can  pay  his  creditors  but  18  J  cents  on  a 
dollar ;  how  much  does  he  owe  ? 

56.  If  8  men  spend  20  dollars  50  cents  in  30  days,  how 
long  will  64  men  be  in  spending  100  dollars  at  the  same 
rate  ? 

57.  A  bridge  built  over  a  stream  in  6  months  by  34  men, 
being  washed  away  by  a  flood,  how  long  time  will  it  take  86 
men  to  build  another  in  its  place,  of  twice  as  much  work  ? 

58.  Three  gardeners,  A,  B,  and  C,  having  bought  a  piece 
of  ground,  find  the  profits  of  it  to  amount  to  240  dollars  a 
year ;  now  the  sum  of  money  which  they  gave,  was  in  such 

R  2 


198  PROMISCUOUS  QUESTIONS. 

proportion,  that  as  often  as  A  paid  5  dolls.  B  paid  7,  and  as 
often  as  B  paid  4  dolls.  C  paid  6 ;  how  much  must  each 
man  receive  for  his  share  of  the  profits  per  annum? 

59.  If  a  county  tax  of  7  cents  and  3  mills  per  cent,  is  as- 
sessed  on  property,  how  much  must  that  man  pay,  whose 
property  is  valued  at  8564  dollars  20  cents  ? 

60.  Suppose  a  cistern  having  a  pipe  which  conveys  4  gal- 
lons 2  quarts  into  it  in  an  hour,  and  has  another  that  lets 
out  2  gallons  2  quarts  and  1  pint  in  an  hour ;  in  what  time 
will  it  be  filled,  allowing  it  to  contain  84^  gallons? 

61.  What  is  the  length  of  a  lane,  which,  being  36  fcul 
wide,  will  contain  just  one  acre  of  ground  ? 

62.  If  50  men  consume  12  bushels  of  grain  in  30  days, 
how  much  will  40  men  consume  in  90  days  ? 

63.  A  gentleman  had  18  dollars  90  cents  to  pay  among 
his  laborers ;  to  every  boy  he  gave  6  cents,  to  every  woman 
S  cents,  and  to  every  man  16  cents;  now  there  were  three 
women  for  every  boy,  and  two  men  for  every  woman  ;  re- 
quired the  number  of  each  ? 

64.  Two  men  depart  from  the  same  place,  and  travel  the 
same  way ;  the  one  travels  at  the  rate  of  3  miles  an  hour, 
for  8  hours  every  day ;  the  other  goes  at  the  rate  of  4J 
miles,  for  7  hours  each  day ;  how  far  are  they  apart  at  the 
end  of  13  days? 

65.  A  began  to  trade  on  the  1st  of  January,  with  a  capi- 
tal of  962  dollars ;  on  the  15th  of  April  following,  he  took 
in  B  as  a  partner,  with  1635  dollars ;  on  the  1st  of  July,  A 
put  in  320  dollars  more,  and  1  month  after  B  drew  out  i 
of  his  capital ;  on  the  last  day  of  December,  on  settling  their 
.accounts,  they  found  a  gain  of  486  dollars  64  cents';  what 

was  each  partner's  share  ? 

66.  Suppose  the  Ohio  river  to  be  2500  feet  wide,  6  feet 
deep,  and  runs  at  the  rate  of  3  miles  an  hour ;  in  what  time 
will  it  fill  a  cistern  of  two  miles  in  length,  breadth,  and 
depth,  the  mile  being  5280  feet? 

67.  A  sloth  was  observed  climbing  a  tree  at  the  rate  of 
9^  inches  every  day,  but  during  the  night  slipped  down  6J 
inches ;  how  long  will  it  be  in  reaching  a  limb  45  feet  6 
inches  from  the  ground! 

68.  In  an  orchard  of  fruit  trees,  £  of  them  bear  applet, 
|  peaches,  £  cherries,  £  plums,  and  46  are  pears;  how 
many  trees  does  the  orchard  contain  ? 

69.  An  old  soldier  lately  received  a  sum  of  money  as 


PROMISCUOUS  QUESTIONS.  199 

pension  from  government :  of  this  sum  he  paid  94  dollars 
in  the  payment  of  debts  which  he  then  owed,  half  of  what 
remained  he  lent  to  a  friend,  and  the  fifth  he  gave  for  a  suit 
of  clothes ;  he  then  found  that  nine-tenths  of  his  money  was 
gone  ;  what  sum  did  he  at  first  receive  ? 

70.  What  number  is  that,  of  which  the  difference  between 
its  third  and  fourth  parts  is  84  ? 

71.  In    turning    a    chaise  within  a  circle   of  a   certain 
diameter,    it  was  discovered  that  the   outer  wheel   turned 
thrice,  while  the  inner  turned  twice;   now  supposing  the 
axle-tree  4  feet  long,  and  the  wheels  of  an  equal  size,  the 
length  of  the  circumference  described  by  each  wheel  is  re- 
quired ? 

72.  The  sum  of  the  sides  of  an  equilateral  triangle  is  125 
feet ;  required  the  area  thereof'/ 

A,  in  a  scuffle,  seized  on  f  of  a  parcel  of  sugar-plums;  B  catched  three- 
eighths  of  it  out  of  his  hands,  and  C  laid  hold  on  three-tenths  more,  D  ran 
off  with  all  that  A  had  left  except  one-seventh,  which  E  afterwards  secured 
slily  for  himself:  then  A  and  C  jointly  set  upon  B,  who  in  the  conflict  let 
fall  £  he  had,  which  was  equally  picked  up  by  D  and  E :  B  then  kicked 
down  C's  hat,  and  to  work  they  went  anew  for  what  it  contained  ;  of  which 
A  got  i,  B  A,  D  two-sevenths,  and  C  and  E  equal  shares  of  what  was  left 
of  that  stock:  D  then  struck  f  of  what  A  and  B  last  acquired  out  of  their 
hands  ;  they  with  difficulty  recoved  five-eighths  of  it  in  equal  shares  again, 
but  the  other  three  carried  off  one -eighth  apiece  of  the  same.  Upon  this 
they  called  a  truce,  and  agreed,  that  the  J-  of  the  whole,  left  by  A  at  first, 
should  be  equally  divided  among  them — How  much  of  the  prize,  after  this 
distribution,  remained  with  each  of  the  competitors  ? 

Ans.  A  got  2863,  B  6335,  C  2438,  D  10294,  and  E  4950. 
Solution. 

First,  f  of  f = J  B's  ^    First  acquisition 
And  y\of  |=i  C's  \    —^  their  sum 

Then  -|— ^=H,  or  ^  left 
I  of  if — J^-  E's  first  acquisition. 
Also,  TV¥ — •£-£o='iTo  D's*    Thus  ended  the  1st  heat. 
Again,  i  of  J  =   J  B's  "j 
Retained  1  C's  !  Part,  at  the   end  of  the 

And    TVo+TV=HtI)'s|  second  scuffle. 

Also  -V3_.  +  Ti_=_i_5_L.  E>s  J 

Proceeding,  i  of  1  =  ^  A's " 

f  of  !=*'! 

Then  ^V  +  rV  +  A=  T^V  to  be  taken 
from  C's.     Thus,  £^-^ 
and  i  of  Tu_  '=j&C9a 

And 


'«>* 

Their  situation 
at  the  end  of 
the  3d  attack. 


200  QUESTIONS  FOR  EXAMINATIONS, 

A.      B. 

Further,  ^  +  ^=J^  and  f  of  &=&  lost  by  A  and  B. 
Then,  tV  of  ft  +  iof  &         ^^i^A's^     p          f 
'  '  " 


Also,   A  of  /,  +  i  of 

And      *  of  W  +  riV  =T*tt*  C's 

i  of  A+W* 

*   Of  T'O  +TWo 

*  Of  *    =    T'? 


_.__l_l_  _  _S_4_3J»  o 

l~3«*0  Tl  5          26880     ^  °    f          nff   ot   jnof 


r  A  got    2863  ^ 


So  that  if  the  }  6335 

sugar-plums  >  then  <j  C  2438  I  Arts. 

were  26880,  )  B  10294 

IE          4950  J 


QUESTIONS  FOR  EXAMINATION. 

THIS  collection  of  questions  is  designed  to  assist  the  teacher  in  l.ie  ex- 
amination of  his  scholars.  It  will  contribute  very  much  to  the  progress  of 
scholars,  to  assign  them  a  certain  number  of  these  questions  as  lessons,  to 
be  answered  correctly  and  with  facility.  Many  similar  questions  will  no 
doubt,  from  time  to  time,  occur  to  the  mind  of  the  teacher,  on  thf  different 
sections,  as  the  scholar  proceeds.  By  accustoming  his  pupils  to  answer 
such  with  ease,  not  only  will  his,  own  burden  in  teaching  be  lei^ened,  but 
the  parents  of  children,  who  have  been  intrusted  to  his  care,  will  find  that 
neither  their  trouble  or  expense  has  been  in  vain. 

PART  I. 

What  is  Arithmetic  ?    How  many  parts  does  it  consist  of? 

What  are  the  characters  used  in  arithmetic  ? 

\Vhat  is  numeration  ?    How  are  the  digits  divided  ? 

What  is  the  rule  for  writing  numbers?    What  is  simple  addition ? 

How  do  you  place  numbers  to  be  added  ?  How  is  the  sum  or  amount  of 
each  column  to  be  set  down  ?  Why  do  you  carry  at  10,  rather  than  for 
any  other  number?  How  is  addition  proved?  What  is  simple  subtrac- 
tion? How  must  the  given  numbers  be  placed  ? 

How  is  subtraction  performed  ?    How  is  subtraction  proved  ? 

What  is  simple  multiplication?    What  are  the  numbers  called? 

In  what  order  are  the  numbers  in  multiplication  to  be  placed  ? 

IIoxv  many  cases  are  there  in  multiplication?  How  is  the  operation  to  be 
performed  in  the  first  and  second  cases?  How  is  multiplication  proved  ? 

When  there  are  ciphers  on  the  right-hand  of  either  of  the  factors,  how  do 
you  proceed  ?  What  is  simple  division  ? 

Wnat  are  the  given  numbers  called  ?    How  are  they  to  be  placed  ? 


QUESTIONS  FOR  EXAMINATION.  201 

How  many  cases  are  there  in  division? 

How  is  division  performed  in  each  case  ?  When  the  multiplier  is  the  exact 
product  of  any  two  factors,  how  do  you  proceed  ? 

When  there  are  ciphers  on  the  right  of  the  divisor,  how  do  you  proceed  ? 

When  the  divisor  is  10,  100,  1000,  &c.  how  do  you  proceed? 

How  is  division  proved  ? 

PART  II. 

Federal  money,  why  so  called  ? 

What  are  its  denominations,  and  standard  weights  ? 

How  is  addition,  subtraction,  multiplication,  and  division  severally  per- 
formed in  federal  money  ?  What  is  compound  addition? 

How  is  compound  addition  performed  \    How  is  it  proved  ? 

What  are  the  denominations  of  English  money,  and  how  are  they  valued  ? 

\Vhat  articles  is  troy  weight  used  for?  What  are  its  denominations  and 
how  valued  ?  What  articles  is  avoirdupois  weight  used  for? 

What  are  its  denominations,  and  how  valued  ?  What  is  apothecaries' 
weight  used  for?  What  are  its  denominations  and  how  valued? 

What  are  the  denominations  of  cloth  measure,  and  how  valued  ? 

What  are  the  denominations  of  long  measure,  and  how  estimated  ? 

What  are  the  denominations  of  land  measure,  and  how  rated  ? 

What  is  cubic  measure,  what  are  its  denominations,  and  relative  difference  ? 

What  are  the  denominations  of  time,  and  what  their  relative  differences  ? 

What  is  the  exact  length  of  the  solar  year? 

What  are  the  denominations  of  motion,  and  the  relative  difference? 

For  what  is  liquid  measure  used,  what  its  denominations,  and  relative  di£ 
ference  ?  What  are  the  denominations  of  dry  measure,  w7hat  used  for 
and  how  estimated  ? 

What  is  compound  subtraction,  and  how  performed  ? 

What  is  compound  multiplication,  how  many  cases,  and  how  performed? 

What  is  compound  division,  how  many  cases,  and  how  performed  ? 

What  is  reduction,  and  how  performed  ?     How  is  reduction  proved  ? 

How  are  pence  reduced  to  cents,  Penn.  currency  ?    How  are  pounds,  shil- 
lings, and  pence,  reduced  to  dollars,  Pennsylvania  currency? 
PART  III. 

What  is  decimal  arithmetic,  and  how  distinguished  from  whole  numbers? 

What  is  the  decimal  point  called? 

What  effect  has  ciphers,  placed  on  the  right-hand  of  the  integer,  and  what 
effect  when  placed  on  the  left-hand  ?  llow  is  addition  of  decimals  per- 
formed ?  How  is  subtraction  of  decimals  performed  ? 

How  is  multiplication  of  decimals  performed  ?     How  is  division  of  decimals 

*    performed  ?    How  many  cases  in  reduction  of  decimals? 

How  is  a  vulgar  fraction  reduced  to  a  decimal  ? 

How  are  numbers  of  different  denominations  reduced  to  a  decimal  of 
equal  value?  How  are  decimals  reduced  to  their  equal  value  in  in- 
tegers ? 

PART  IV. 

What  is  proportion  ?    Into  how  many  parts  is  it  divided  ? 

What  are  the  given  terms  in  proportion  called  ? 

Whaf  is  required  in  the  single  rule  of  three  direct  ? 

How  may  you  know  when  the  question  is  in  direct  proportion? 

What  is  the  rule  for  stating  questions  in  the  single  rule  of  three  direct? 

How  is  the  operation  performed  ?  How  do  you  prove  questions  in  the  single 
rule  of  three  direct  ?  What  is  the  single  rule  of  three  inverse  ? 

How  may  you  know  when  the  question  is  in  the  single  rule  of  three  inverse  ? 

How  is  the  operation  performed  in  the  single  rule  of  three  inverse  ?  How 
are  questions  proved  in  this  rule  ?  What  is  the  double  rule  of  three  ? 

How  many,  and  which  terms  must  be  a  supposition,  and  how  many,  and 
which  must  be  a  demand  ? 

What  is  the  rule  for  stating  questions  in  the  double  rule  of  three  ? 

[low  is  the  operation  performed  in  the  double  rule  of  three  direct? 


202  QUESTIONS  FOR  EXAMINATION. 

How  do  you  know  when  the  question  is  in  direct  proportion,  and  when  in 
inverse?    How  is  the  operation  performed  in  inverse  proportion  ? 
PART  V. 

What  is  practice,  and  why  so  called  ?  How  many  cases  are  there  in  practice  ? 

When  the  price  consists  of  dollars,  cents,  ana  mills,  how  is  the  operation 
performed  ?  When  the  price  is  the  fractional  part  of  a  dollar  or  cent, 
how  is  the  operation  performed  ?  When  the  price  and  quantity  given 
are  of  several  denominations,  how  is  the  operation  performed  ? 

When  the  price  consists  of  pounds,  shillings,  pence,  and  farthings,  how  do 
you  proceed  ?  What  is  meant  by  aliquot  parts? 

When  both  the  price  of  the  integer  and  the  quantity  are  of  different  de- 
nominations, how  do  you  proceed  ? 

What  is  tare  and  tret,  and  what  is  gross  and  neat? 

How  do  you  work  questions  in  tare  and  tret?     What  is  interest? 

What  is  the  general  rate  of  interest?  What  is  the  sum  of  money  loaned, 
called?  What  do  you  understand  by  the  amount? 

How  many  kinds  of  interest  are  there  ?     What  is  simple  interest  ? 

How  many  cases  are  there  in  simple  interest?  When  the  given  time  is 
years  and  the  principal  dollars,  how  is  the  interest  found  ? 

When  there  are  cents  and  mills  in  the  principal,  how  do  you  proceed  ? 

When  the  lime  is  years  and  months,  or  months  *only,  how  is  the  interest 
found  ?  When  the  time  is  months  and  days,  or  days  only,  how  is  the  ope- 
ration performed?  How  is  the  interest  computed  on  bonds,  notes,  &C;? 

What  is  compound  interest,  and  how  is  it  performed  ?    What  is  insurance  ? 

What  is  the  instrument  of  agreement  termed  ? 

How  are  the  questions  in  insurance  performed  ? 

What  is  commission,  and  how  performed  ?  What  is  brokage,  and  how  per- 
formed ?  What  is  stock,  and  how  bought  and  sold  ? 

What  is  rebate  or  discount,  and  what  the  rule  to  work  questions  therein  ? 

What  is  the  difference  between  discount  and  interest? 

What  is  bank  discount,  and  how  is  the  discount  calculated  ? 

What  do  you  mean  by  the  equation  of  payments? 

How  is  the  mean  time  found  in  the  equation  of  payments  ? 

What  is  fellowship,  and  how  many  kinds  are  there  ? 

What  is  single  fellowship,  and  how  is  the  operation  performed  ? 

What  is  compound  fellowship,  and  what  is  the  rule  for  working  questions 
therein  ?  What  is  profit  and  loss,  and  what  is  the  rule  of  operation 
therein  ?  What  is  barter,  and  how  performed  ? 

What  is  exchange,  ami  of  how  many  kinds  ? 

What  do  you  understand  by  par  in  exchange,  and  what  by  agio? 

How  do  you  reduce  the  currency  of  different  states  to  federal  money? 

How  do  you  reduce  the  currency  of  one  state  to  another,  where  it  is  differ- 
ent in  them?  How  are  accounts  kept  in  England,  Ireland,  and  how  in 
France,  Spain,  &c.  ?  What  is  alligation,  and  how  many  cases  are  therein  ? 

How  are  the  operations  performed  in  the  first  case,  second  case,  &c.  ? 
PART  VI. 

What  is  a  vulgar  fraction,  and  how  many  kinds  are  there  ? 

What  is  a  proper  fraction,  what  is  an  improper  fraction,  what  is  a  com- 
pound fraction,  and  what  is  a  mixed  fraction? 

What  are  the  numbers  above  the  fine  called,  and  also  those  below? 

How  do  you  reduce  vulgar  fractions  to  their  lowest  terms  ? 

EIow  are  mixed  numbers  reduced  to  an  improper  fraction  ? 

How  is  an  improper  fraction  reduced  to  a  whole  or  mixed  number? 

How  do  you  reduce  fractions  to  others  that  shall  have  a  common  denomina- 
tor? How  do  you  find  the  least  common  denominator? 

How  do  you  find  the  value  of  a  fraction,  in  the  known  parts  of  an  integer? 

How  are  given  quantities  reduced  to  the  fraction  of  a  greater  denomination  ? 

How  are  vulgar  fractions  reduced  to  decimals  of  the  same  value  ? 

How  do  you  reduce  a  compound  fraction  to  a  single  one? 

How  are  vulgar  fractions  added,  subtracted,  multiplied,  and  divided? 


QUESTIONS  FOR  EXAMINATION.  203 

How  do  you  perform  the  single  rule  of  three  in  vulgar  fractions,  direct, 
and  inverse  ? 

PART  VII. 

What  is  involution  ?    What  do  you  understand  by  the  power  of  a  number  ? 

ilow  is  involution  performed  ?  What  is  the  number  denoting  the  power, 
termed  ?  What  is  evolution  ?  What  do  you  understand  by  a  root  ? 

How  do  you  extract  the  square  root  ? 

If  there  be  decimals  in  the  given  number,  how  must  it  be  pointed? 

How  do  you  extract  the  square  root  of  a  vulgar  fraction? 

How  is  the  square  root  of  a  mixed  number  extracted  ? 

Hew  do  you  find  the  side  of  a  right  angled  triangle,  the  other  two  being 
given  f  How  do  you  find  the  side  of  a  square,  in  any  given  area  ? 

How  do  you  find  the  diameter  of  a  circle,  when  the  area  is  given  ? 

How  do  you  prove  the  square  root?    What  is  a  cube ? 

How  do  you  extract  the  cube  root  in  whole  numbers? 

Ilow  do  you  extract  the  cube  of  a  vulgar  fraction  ? 

How  do  you  point  off  in  decimal  numbers?  How  is  the  cube  root  of  a 
mixed  number  extracted  ?  How  is  the  cube  root  proved  ? 

What  is  progression,  and  how  many  kinds  ?  What  is  principally  to  be  ob- 
served in  arithmetical  progression  ?  How7  do  you  find  the  last  term ;  and 
sum  of  all  the  terms  ?  How  do  you  find  the  common  difference  ? 

vVhat  is  geometrical  progression,  and  how  does  it  differ  from  arithmetical  ? 

What  is  principally  to  be  observed  in  geometrical  progression  ? 

How  do  you  find  the  last  term,  and  sum  of  all  the  series  ? 

What  is  position,  and  of  how  many  kinds  ?  How  do  you  resolve  questions 
in  singlo  position?  How  is  single  position  proved  ?  What  is  double  po- 
sition ?  What  is  the  rule  for  working  questions  in  double  position  ? 

What  do  you  understand  by  permutation  ?  How  is  the  number  of  varia- 
tions found  in  this  rule?  What  do  we  learn  from  the  results  of  this  rule  ? 

What  is  combination?  How  do  you  find  the  greatest  possible  number  of 
combinations  in  any  given  number? 

PART  Vlli. 

VVhat  are  duodecimals?  What  are  the  denominations  in  duodecimals,  and 
what  are  they  termed?  How  are  duodecimals  added,  subtracted,  and 
multiplied  ?  How  do  you  prove  multiplication  of  duodecimals  ? 

How  is  the  solid  content  of  bales,  &c.  found  by  duodecimals? 

How  do  you  find  a  ship's  tonnage  ?  What  is  the  carpenters'  rule,  and  what 
its  use  ?  How  do  you  find  the  superficial  content  of  boards,  &c.  ? 

How  do  you  find  the  solid  content  of  squared  timber? 

How  do  you  find  the  solid  content  of  round  timber? 

What  things  belong  to  carpenters'  work?  By  what  numbers  do  carpenters 
usually  measure  their  work  ?  How  is  brick- work  estimated  ? 

What  is  the  standard  thickness  of  a  brick  wall? 

How  do  you  reduce  a  wall  of  a  different  thickness  to  a  standard  one  ? 

How  is  masons'  work  measured  ?  What  kind  of  measure  is  used  for  ma- 
terials? How  is  the  solid  content  of  walls  calculated  ? 

What  is  superficial  measure  ?    How  is  plasterers'  work  divided  ? 

In  what  manner  is  plasterers'  work  measured  ?  How  is  white-washing 
and  coloring  estimated  ?  How  is  pavers'  work  calculated  ? 

How  do  painters  compute  their  work  ?  In  what  manner  is  glaziers'  work 
estimated  ?  How  are  the  contents  of  squares  calculated? 

In  what  way  is  the  area  of  an  oblong  piece  of  ground  ascertained  ? 

How  do  you  calculate  triangular  pieces  of  ground  ? 

How  do  you  calculate  a  piece  of  ground  lying  in  the  shape  of  an  obiiquo 
parallelogram?  How  are  pieces  of  ground,  oounded  by  four  irregular 
sides,  calculated,?  How  do  you  calculate  the  area  of  a  circle? 

What  is  gauging?     How  are  the  contents  of  casks,  &c.  calculated? 

How  do  you  calculate  the  power  of  the  lever? — of  the  axle  and  wheel  ? 

VVhat  things  are  to  be  considered  in  finding  the  power  of  the  screw? 

How  do  you  find  the  proportion  between  the  weight  and  the  power? 


CONTENTS. 


PART  I. 

P^?e 

Numeration 7 

Addition    9 

Subtraction 11 

Multiplication 13 

Division    17 

PART  II. 

Federal  Money 21 

Compound  Addition 24 

Compound  Subtraction 30 

Compound  Multiplication   ....  32 

Compound  Division 35 

Reduction  39 

PART  III. 

Addition  of  Decimals 46 

Subtraction  of  Decimals   47 

Multiplication  of  Decimals  ...  48 

Division  of  Decimals 50 

Reduction  of  Decimals   52 

PART  IV. 

Single  Rule  of  Three  Direct  . .  56 
Single  Rule  of  Three  Inverse  .  59 
Double  Rule  of  Three 65 

PART  V. 

Practice 68 

Tare  and  Tret 73 

Simple  Interest 75 

Compound  Interest 85 

Insurance,  Commission,  and 

Brokage 87 

Buying  and  Selling  Stocks   . .    89 

Rebate  or  Discount    90 

Bank  Discount 92 

Equation  of  Payments 93 

Single  Fellowship 95 

Compound  Fellowship 97 

Profit  and  Loss 99 

Barter   101 

Exchange 102 

Alligation   109 


PART  VI. 

}•,-.?< 

Vulgar  Fractions    113 

Reduction  of  Vulgar  Frac- 
tions   '. ib. 

Addition  of  Vulgar  Fractions    120 
Subtraction  of  Vulgar  Fr<ic- 

tions ..;  .  122 

Multiplication  of  Vulgar  Frac- 
tions     123 

Division  of  Vulgar  Frac- 
tions   124 

The  Rule  of  Three  in  Vulgar 

Fractions 125 

Inverse  Proportion 12G 

PART  VII. 

Involution 127 

Evolution 128 

The  Square  Root    ib. 

The  Cube  Root 137 

Arithmetical  Progression  ....  141 

Geometrical  Progression   ....  144 

Single  Position   147 

Double  Position 148 

Permutation    151 

Combination   152 

PART  VIII. 

Duodecimals  153 

The  Carpenters'  Rule    159 

Measuring  of  Boards  and 

Timber    162 

Carpenters'  and  Joiners'  Work  166 

Bricklayers'  Work   170 

Masons' Work    174 

Plasterers'  Work    176 

Pavers' Work   177 

Painters'  Work 178 

Glaziers' Work 180 

Measurement  of  Ground 181 

Gauging    188 

Mechanical  Powers   190 

Promiscuous  Questions 192 

Questions  for  Examination  . .  200 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPT. 

This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 
Renewed. books  are  subject  tn  imnnpHmtp  recall. 


ICLf 


JAN  2  01968  6  5 


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